New generalized Pólya–Szegö and Čebyšev type inequalities with general kernel and measure

Abstract

It is always attractive and motivating to acquire the generalizations of known results. In this article, we introduce a new class $\mathfrak{C(h)}$ of functions which can be represented in a form of integral transforms involving general kernel with σ-finite measure. We obtain some new Pólya–Szegö and Čebyšev type inequalities as generalizations to the previously proved ones for different fractional integrals including fractional integral of a function with respect to another function capturing Riemann–Liouville integrals, Hadamard fractional integrals, Katugampola fractional integral operators, and conformable fractional integrals. This new idea shall motivate the researchers to prove the results over a measure space with general kernels instead of special kernels.

1 Introduction

There are several problems in the mathematics and its related real world applications wherein fractional derivatives occupy an important place [1–13]. In many technologies, the fractional derivatives take a place in a way they can be described in different approaches, where these approaches can be used to explain a lot of essential real world problems. Each conventional fractional operator with its own special kernel can be used in a certain problem. Analyzing the uniqueness of fractional ordinary and partial differential equations can be performed by employing fractional integral inequalities. In the literature many applications can be found (for example, see [14–20]). For broader applications, we recommend the reader to see [21–29]. Recently, many authors have utilized unique versions of such inequalities to study diverse classes of differential and integral equations. Such types of inequalities are considered as far-reaching tools that demonstrate the analytical properties of several classes of differential and integral equations [30–41]. Čebyšev [42] came up with his well-known celebrated functional as follows:

For Lebesgue integrable functions $\chi _{1},\chi _{2} :[a_{1},a_{2}] \to {\mathbb{R}}$, we consider the Čebyšev functional

$$ \begin{aligned}[b] \Omega (\chi _{1},\chi _{2})&= \frac{1}{a_{2}-a_{1}} \int _{a_{1}}^{a_{2}} \chi _{1}(\rho )\chi _{2}(\rho )\,d\rho \\ &\quad {}-\frac{1}{a_{2}-a_{1}} \int _{a_{1}}^{a_{2}} \chi _{1}(\rho )\,d\rho \cdot \frac{1}{a_{2}-a_{1}} \int _{a_{1}}^{a_{2}} \chi _{2}(\rho )\,d \rho , \end{aligned} $$

(1.1)

where $\chi _{1}$ and $\chi _{2}$ are two integrable functions on $[a_{1},a_{2}]$. If $\chi _{1}$ and $\chi _{2}$ are synchronous, that is,

$$ \bigl[\chi _{1}(\zeta )-\chi _{1}(\varsigma )\bigr] \bigl[ \chi _{2}(\zeta )-\chi _{2}( \varsigma )\bigr]\geq 0 $$

for any $\gamma ,\omega \in [a_{1},a_{2}]$, then $\Omega (\chi _{1},\chi _{2})\geq 0$.

An enormous amount of efforts have been devoted to sharpening and extension of the classical inequalities. The inequality presented in the following definition is one of the interesting parts of the theory of classical inequalities known as Grüss type inequality [35]

$$ \bigl\vert \Omega (\chi _{1},\chi _{2}) \bigr\vert \leq \frac{(C-c)(D-d)}{4}, $$

where $\chi _{1}$ and $\chi _{2}$ are integrable functions that satisfy

$$ c \leq \chi _{1}(\zeta ) \leq C $$

and

$$ d \leq \chi _{2}(\zeta ) \leq D $$

for all $\gamma \in [a_{1}, a_{2}]$ and for some $c,C,d,D \in \mathbb{R}$.

Pólya–Szegö introduced one of the most intensively studied inequalities in [43] stated as follows:

$$ \frac{\int _{a_{1}}^{a_{2}} \chi _{1}^{2}(\zeta )\,d\zeta \int _{a_{1}}^{a_{2}} \chi _{2}^{2}(\zeta )\,d\zeta }{\int _{a_{1}}^{a_{2}} \chi _{1}(\zeta )\chi _{2}(\zeta )\,d\zeta } \leq \frac{1}{4} \biggl(\sqrt{ \frac{cd}{CD}}+\sqrt{\frac{CD}{cd}} \biggr)^{2}. $$

(1.2)

The constant $\frac{1}{4}$ is least possible such that inequality (1.2) is valid.

Dragomir and Diamond in [44] proved the following Grüss type inequality via inequality (1.2) by utilizing the Pólya–Szegö inequality:

$$ \bigl\vert \Omega (\chi _{1},\chi _{2}) \bigr\vert \leq \frac{(C-c)(D-d)}{4(a_{2}-a_{1})\sqrt{cdCD}\int _{a_{1}}^{a_{2}} \chi _{1}^{2}(\zeta )\,d\zeta \int _{a_{1}}^{a_{2}} \chi _{2}^{2}(\zeta )\,d\zeta }. $$

Let $(\Delta , \Sigma ,\beta )$ be measure space with positive σ-finite measure, $h: \Delta \times \Delta \to {\mathbb{R}}$ be a nonnegative function, and

$$ \Upsilon (\varsigma )= \int _{\Delta } h(\varsigma ,\zeta ) \,d \beta (\zeta ) , \quad \varsigma \in \Delta . $$

(1.3)

Throughout this paper, we suppose $\Upsilon (\varsigma )>0$ a.e. on Δ.

Let $\mathfrak{C(h)}$ denote the class of functions $\mho : \Delta \to {\mathbb{R}}$ with the representation

$$ \overline{\mho }(\varsigma )= \int _{\Delta }h(\varsigma ,\zeta ) \mho (\zeta )\,d\beta (\zeta ), $$

where $\overline{\mho }:\Delta \rightarrow \mathbb{R}$ is a measurable function.

Definition 1.1

(see [13, 45])

Let $\vartheta \in L_{1}[a,b]$ and $\mathfrak{R}_{a^{+}}^{\varrho }\vartheta$ and $\mathfrak{R}_{b^{-}}^{\varrho }\vartheta$ be the left-sided and right-sided Riemann–Liouville fractional integrals of order $\varrho >0$ defined by

$$ \mathfrak{R}_{a^{+}}^{\varrho }\vartheta (\varsigma )= \frac{1}{\Gamma (\varrho )} \int _{a}^{\varsigma }\vartheta ( \zeta ) (\varsigma -\zeta )^{\varrho -1}\,d\zeta \quad (\varsigma >a) $$

and

$$ \mathfrak{R}_{b^{-}}^{\varrho }\vartheta (\varsigma )= \frac{1}{\Gamma (\varrho )} \int _{\varsigma }^{b}\vartheta ( \zeta ) (\zeta -\varsigma )^{\varrho -1}\,d\zeta \quad (\varsigma < b), $$

respectively, where $\Gamma (\varrho )=\int _{0}^{\infty }e^{-\wp }{-\wp }^{\varrho -1}\,d \wp$ is the usual gamma function.

Diaz et al. in [46] originated the following definition of gamma k-function.

Definition 1.2

The generalized Γ function known as $\Gamma -k$ function is defined by the following definition:

$$ \Gamma _{k}(\zeta )=\lim_{m\rightarrow \infty } \frac{m!k^{m}(mk)^{\frac{t}{k}-1}}{(\zeta )_{m,k}}, \quad k>0, \mathbb{R}(\zeta )>0, $$

where $(\zeta )_{m,k}=t(\zeta +k)(\zeta +2k)\ldots$, $(\zeta +(m-1)k)$, $m \geq 1$, is Pochhammer k symbol. The generalized gamma function can also be written as

$$ \Gamma _{k}(\zeta )= \int _{0}^{\infty }x^{\zeta -1}e^{ \frac{-x^{k}}{k}}\,dx, \quad \mathbb{R}(\zeta )>0. $$

(1.4)

Specially, for $k=1$, $\Gamma _{1}(\zeta )=\Gamma (\zeta )$.

Definition 1.3

([47])

Let $f\in L_{1}([a,b])$ (the Lebesgue measure). The left-sided and right-sided Riemann–Liouville k-fractional integrals $\mathfrak{R}_{a^{+}}^{\varrho ,k}f$ and $\mathfrak{R}_{b^{-}}^{\varrho ,k}f$ of order $\varrho >0$ are defined by

$$ \mathfrak{R}_{a^{+}}^{\varrho ,k}f(\varsigma )= \frac{1}{k\Gamma _{k}(\varrho )} \int _{a}^{\varsigma }f(\zeta ) ( \varsigma -\zeta )^{\frac{\varrho }{k}-1}\,d\zeta \quad (\varsigma >a) $$

and

$$ \mathfrak{R}_{b^{-}}^{\varrho ,k}f(\varsigma )= \frac{1}{k\Gamma _{k}(\varrho )} \int _{\varsigma }^{b}f(\zeta ) ( \zeta -\varsigma )^{\frac{\varrho }{k}-1}\,d\zeta \quad (\varsigma < b), $$

where $\Gamma _{k}(\cdot)$ is the k-gamma function.

In 2006 Kilbas et al. in [48] presented the definition of fractional integrals concerning another function as follows.

Definition 1.4

Let $(a,b)$, $-\infty \leq a < b \leq \infty$, and $\varrho >0$ ψ be a positive increasing function on $(a,b]$. The left-sided and right-sided fractional integrals of a function f concerning another function ψ in $[a,b]$ are given by

$$ I_{a+;\psi }^{\varrho }f(\varsigma )=\frac{1}{\Gamma (\varrho )} \int _{a}^{\varsigma }\frac{\psi '(\zeta )f(\zeta )\,d\zeta }{[\psi (\varsigma )-\psi (\zeta )]^{1-\varrho }}, \quad \varsigma >a, $$

and

$$ I_{b-;\psi }^{\varrho }f(\varsigma )=\frac{1}{\Gamma (\varrho )} \int _{x}^{b} \frac{\psi '(\zeta )f(\zeta )\,d\zeta }{[\psi (\zeta )-\psi (\varsigma )]^{1-\varrho }}, \quad \varsigma < b. $$

The classical Hadamard fractional integral and its generalized form is given in the next two definitions.

Definition 1.5

Let $(a,b)$ be a finite or infinite interval of the half axis $\mathbb{R}_{+}$ and $\varrho >0$. The Hadamard-type fractional integrals of order $\varrho >0$ are given by

$$ J_{a_{+}}^{\varrho }f(\varsigma )=\frac{1}{\Gamma (\varrho )} \int _{a}^{\varsigma } \biggl(\log \frac{\varsigma }{\zeta } \biggr)^{\varrho -1}\frac{f(\zeta )\,d\zeta }{\zeta },\quad \varsigma >a, $$

and

$$ J_{b_{-}}^{\varrho }f(\varsigma )=\frac{1}{\Gamma (\varrho )} \int _{x}^{b} \biggl(\log \frac{\zeta }{\varsigma } \biggr)^{ \varrho -1}\frac{f(\zeta )\,d\zeta }{\zeta },\quad \varsigma < b, $$

respectively.

The generalized Hadamard type fractional integrals are stated in the next definition.

Definition 1.6

Let $(a,b)$ be a finite or infinite interval of the half axis $\mathbb{R}_{+}$ and $\varrho >0$. The Hadamard fractional integrals of order $\varrho >0$ are given by

$$ J_{a_{+}}^{\varrho ,k}f(\varsigma )=\frac{1}{\Gamma _{k}(\varrho )} \int _{a}^{\varsigma } \biggl(\log \frac{\varsigma }{\zeta } \biggr)^{\frac{\varrho }{k}-1} \frac{f(\zeta )\,d\zeta }{y},\quad \varsigma >a, $$

and

$$ J_{b_{-}}^{\varrho ,k}f(\varsigma )=\frac{1}{\Gamma _{k}(\varrho )} \int _{x}^{b} \biggl(\log \frac{y}{x} \biggr)^{ \frac{\varrho }{k}-1}\frac{f(\zeta )\,d\zeta }{y},\quad \varsigma < b, $$

respectively.

Recently, Saima et al. in [49] generalized the definition of operator given in Definition 1.4 stated as follows.

Definition 1.7

Let $(a,b)$ be a finite or infinite interval on the real line ℜ together with $k>0$. Let $\psi >0$ be an increasing function on $(a,b]$, then the generalized fractional integrals of a function f concerning another function ψ of order $\varrho >0$ are given by

$$ I_{a+;\psi }^{\varrho ,k}f(\varsigma )=\frac{1}{\Gamma _{k}(\varrho )} \int _{a}^{\varsigma }\frac{\psi '(\zeta )f(\zeta )\,d\rho }{[\psi (\varsigma )-\psi (\zeta )]^{1-\frac{\varrho }{k}}}, \quad \varsigma >a, $$

and

$$ I_{b-;\psi }^{\varrho ,k}f(\varsigma )=\frac{1}{\Gamma (\varrho )} \int _{x}^{b} \frac{\psi '(\zeta )f(\zeta )\,d\rho }{[\psi (\zeta )-\psi (\varsigma )]^{1-\frac{\varrho }{k}}}, \quad \varsigma < b. $$

Note that corresponding to $\psi (\varsigma )=\log (\varsigma )$ the above defined operator represents the Hadamard fractional integrals of order ϱ.

The definition of the Erdélyi–Köber fractional integrals is given by the following definition. For details, we refer the reader to the book [50].

Definition 1.8

Let $(a, b)$ be a finite or infinite interval of the half axis $\mathbb{R}_{+}$ together with $\varrho , \sigma > 0$ and $\eta \in {\mathbb{R}}$. The left-sided and right-sided integrals of order $\varrho \in {\mathbb{R}}$ are defined by

$$ I_{a_{+};\sigma ;\eta }^{\varrho }f(\varsigma )= \frac{\sigma \varsigma ^{-\sigma (\varrho +\eta )}}{\Gamma (\varrho )} \int _{a}^{\varsigma }\frac{\zeta ^{\sigma \eta +\sigma -1}f(\zeta )\,d\zeta }{(\varsigma ^{\sigma }-\zeta ^{\sigma })^{1-\varrho }} $$

(1.5)

and

$$ I_{b_{-};\sigma ;\eta }^{\varrho }f(\varsigma )= \frac{\sigma \varsigma ^{\sigma \eta }}{\Gamma (\varrho )} \int _{\varsigma }^{b} \frac{zeta^{\sigma (1-\eta -\varrho )-1} f(\zeta )\,d\zeta }{(\zeta ^{\sigma }-\zeta ^{\sigma })^{1-\varrho }}, $$

(1.6)

respectively.

Consider the space $X_{c}^{p}(a,b)$, ($c\in \mathbb{R}$, $1\leq p\leq \infty$) of complex-valued Lebesgue measurable functions f on $[a,b]$ such that $\|f\|_{X_{c}^{p}(a,b)}<\infty$, where

$$ \Vert f \Vert _{X_{c}^{p}}= \biggl( \int _{a}^{b} \bigl\vert \varsigma ^{c}f( \varsigma ) \bigr\vert ^{p}\frac{d\varsigma }{\varsigma } \biggr)< \infty . $$

Definition 1.9

Let $[a, b]\subset \mathbb{R}$ be a finite interval. Then the left- and right-sided Katugampola fractional integrals of order $\varrho >0$ of $f\in X_{c}^{p}(a,b)$ are defined by

$$ {}^{\rho }I_{a_{+}}^{\varrho }f(\varsigma )= \frac{\rho ^{1-\varrho }}{\Gamma (\varrho )} \int _{a}^{\varsigma }\frac{\zeta ^{\rho -1}f(\zeta )\,d\zeta }{(\varsigma ^{\rho }-\zeta ^{\rho })^{1-\varrho }} $$

(1.7)

and

$$ {}^{\rho }I_{b_{-}}^{\varrho }f(\varsigma )= \frac{\rho ^{1-\varrho }}{\Gamma (\varrho )} \int _{\varsigma }^{b} \frac{\zeta ^{\rho -1}f(\zeta )\,d\zeta }{(\zeta ^{\rho }-\varsigma ^{\rho })^{1-\varrho }} $$

(1.8)

with $a < nu < b$ and $\rho >0$, if the integral exists.

Definition 1.10

([51])

Let $\beta \in \mathbb{C}$, $\mathbb{R}(\beta )>0$. We define the left-fractional conformable integral operator by

$$ {}_{\beta }^{\rho }\mathfrak{J}^{\varrho }f( \varsigma )= \frac{1}{\Gamma (\beta )} \int _{a}^{\varsigma } \biggl( \frac{(\varsigma -a)^{\varrho }-(\zeta -a)^{\varrho }}{\varrho } \biggr)^{ \beta -1}f(\zeta )\frac{d\zeta }{(\zeta -a)^{1-\varrho }} $$

(1.9)

and

$$ {}_{\beta }^{\rho }\mathfrak{J}^{\varrho }f( \varsigma )= \frac{1}{\Gamma (\beta )} \int _{\varsigma }^{b} \biggl( \frac{(b-\varsigma )^{\varrho }-(b-\zeta )^{\varrho }}{\varrho } \biggr)^{ \beta -1}f(\zeta )\frac{d\zeta }{(b-\zeta )^{1-\varrho }}. $$

(1.10)

Definition 1.11

([52])

Let ϕ be a confirmable fractional integral on the interval $[p,q]\subseteq (0,\infty )$. The right-sided and left-sided generalized conformable fractional integrals ${}_{\varrho }^{\tau }K_{p^{+}}^{\beta }$ and ${}_{\varrho }^{\tau }K_{q^{-}}^{\beta }$ of order $\beta >0$, $\tau \in \mathbb{R}$, $\varrho +\tau \neq 0$, are defined by

$$ {}_{\varrho }^{\tau }K_{p^{+}}^{\beta } \phi (\zeta )= \frac{1}{\Gamma (\beta )} \int _{p}^{\zeta } \biggl( \frac{\zeta ^{\varrho +\tau }-\varsigma ^{\varrho +\tau }}{\varrho +\tau } \biggr)^{\beta -1}\phi (\varsigma )\varsigma ^{\tau }d_{\varrho } \varsigma $$

(1.11)

and

$$ {}_{\varrho }^{\tau }K_{q_{-}}^{\beta } \phi (\zeta )= \frac{1}{\Gamma (\beta )} \int _{\zeta }^{q} \biggl( \frac{\varsigma ^{\varrho +\tau }-\zeta ^{\varrho +\tau }}{\varrho +\tau } \biggr)^{\beta -1}\phi (\varsigma )\varsigma ^{\tau }d_{\varrho } \varsigma , $$

(1.12)

respectively, ${}_{\varrho }^{\tau }K_{p^{+}}^{0}\phi (\zeta )= {{}_{\varrho }^{\tau }}K_{q_{-}}^{0} \phi (\zeta )=\phi (\zeta )$.

2 Main results

In the present section, we set up the Pólya–Szegö and Cebyšev type inequalities for the general kernel with related applications in fractional calculus.

Theorem 2.1

Let $(\Delta , \Sigma ,\beta )$ be a measure space with positive σ-finite measure. Let $h: \Delta \times \Delta \to {\mathbb{R}}$ be nonnegative and $\chi _{1}, \chi _{2}, \Upsilon _{1}, \Upsilon _{2}, \Omega _{1}, \Omega _{2}\in \mathfrak{C(h)}$ be positive integrable functions defined on $[0,\zeta )$ such that

$$\begin{aligned} 0< \Upsilon _{1}(\varsigma )\leq \chi _{1}( \varsigma )\leq \Upsilon _{2}( \varsigma ),\qquad 0< \Omega _{1}(\varsigma )\leq \chi _{2}(\varsigma ) \leq \Omega _{2}(\varsigma ) \end{aligned}$$

(2.1)

for all $\varsigma \in [0,\infty )$. Then

$$\begin{aligned} & \bigl( \overline{\bigl(\Upsilon _{1}(\zeta )\Omega _{1}( \zeta )+\Upsilon _{2}(\zeta )\Omega _{2}(\zeta )\bigr)\chi _{1}(\zeta )\chi _{2}(\zeta )} \bigr)^{2} \\ &\quad \geq 4 \overline{\Omega _{1}(\zeta )\Omega _{2}( \zeta )\chi _{1}^{2}(\zeta )} \cdot \overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )\chi _{2}^{2}( \zeta )}. \end{aligned}$$

(2.2)

Proof

It follows from (2.1) that

$$ \biggl(\frac{\Upsilon _{2}(\varsigma )}{\Omega _{1}(\varsigma )}- \frac{\chi _{1}(\varsigma )}{\chi _{2}(\varsigma )} \biggr)\geq 0 $$

(2.3)

and

$$ \biggl(\frac{\chi _{1}(\varsigma )}{\chi _{2}(\varsigma )}- \frac{\Upsilon _{1}(\varsigma )}{\Omega _{2}(\varsigma )} \biggr) \geq 0. $$

(2.4)

Multiplying inequalities (2.3) and (2.4), we get

$$ \begin{aligned}[b] &\bigl[\Upsilon _{1}(\varsigma )\Omega _{1}( \varsigma )+\Upsilon _{2}( \varsigma )\Omega _{2}(\varsigma ) \bigr]\chi _{1}(\varsigma )\chi _{2}( \varsigma )\\ &\quad \geq \Omega _{1}(\varsigma )\Omega _{2}(\varsigma )\chi _{1}^{2}( \varsigma )+\Upsilon _{1}(\varsigma ) \Upsilon _{2}(\varsigma )\chi _{2}^{2}( \varsigma ). \end{aligned} $$

(2.5)

Multiplying by $h(\zeta ,\varsigma )$ and integrating with respect to ς over measure space Δ, we get that

$$\begin{aligned} & \int _{\Delta }h(\zeta ,\varsigma )\bigl[\Upsilon _{1}( \varsigma ) \Omega _{1}(\varsigma )+\Upsilon _{2}(\varsigma ) \Omega _{2}( \varsigma )\bigr]\chi _{1}(\varsigma )\chi _{2}(\varsigma )\,d\beta ( \varsigma ) \\ &\quad \geq \int _{\Delta }h(\zeta ,\varsigma )\Omega _{1}( \varsigma )\Omega _{2}(\varsigma )\chi _{1}^{2}(\varsigma ) \,d\beta ( \varsigma ) + \int _{\Delta }h(\zeta ,\varsigma )\Upsilon _{1}( \varsigma )\Upsilon _{2}(\varsigma )\chi _{2}^{2}( \varsigma )\,d\beta ( \varsigma ). \end{aligned}$$

Using the definition of $\mathfrak{C(h)}$, we can write

$$\begin{aligned} &\overline{\bigl[\Upsilon _{1}(\zeta )\Omega _{1}(\zeta )+ \Upsilon _{2}(\zeta )\Omega _{2}(\zeta )\bigr]\chi _{1}(\zeta )\chi _{2}(\zeta )} \\ &\quad \geq \overline{\Omega _{1}(\zeta )\Omega _{2}(\zeta )\chi _{1}^{2}(\zeta )}+ \overline{\Upsilon _{1}( \zeta )\Upsilon _{2}(\zeta )\chi _{2}^{2}(\zeta )}. \end{aligned}$$

(2.6)

Now applying the arithmetic-geometric inequality, we have

$$\begin{aligned} &\overline{\bigl[\Upsilon _{1}(\zeta )\Omega _{1}(\zeta )+ \Upsilon _{2}(\zeta )\Omega _{2}(\zeta )\bigr]\chi _{1}(\zeta )\chi _{2}(\zeta )} \\ &\quad \geq 2\sqrt{ \overline{\Omega _{1}(\zeta )\Omega _{2}(\zeta )\chi _{1}^{2}(\zeta )} \,\overline{ \Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )\chi _{2}^{2}(\zeta )}} , \end{aligned}$$

(2.7)

which leads to

$$\begin{aligned} &\frac{1}{4}\bigl( \overline{\bigl[\Upsilon _{1}(\zeta )\Omega _{1}(\zeta )+\Upsilon _{2}(\zeta )\Omega _{2}( \zeta )\bigr]\chi _{1}(\zeta )\chi _{2}(\zeta )} \bigr)^{2} \\ &\quad \geq \overline{\Omega _{1}(\zeta )\Omega _{2}(\zeta )\chi _{1}^{2}(\zeta )} \,\overline{\Upsilon _{1}( \zeta )\Upsilon _{2}(\zeta )\chi _{2}^{2}(\zeta )}, \end{aligned}$$

(2.8)

which implies (2.2). □

Corollary 2.2

Let $\zeta , c,d,C,D>0$ with $q \leq Q$, $r \leq R$, and let $\chi _{1}$, $\chi _{2}$ be two positive integrable functions defined on $[0,\infty )$ such that

$$\begin{aligned} 0< c\leq \chi _{1}(\varsigma )\leq C< \infty ,\qquad 0< d \leq \chi _{2}( \varsigma )\leq D< \infty \end{aligned}$$

(2.9)

for all $\varsigma \in [0,\infty )$. Then

$$\begin{aligned} \frac{\overline{\chi _{1}^{2}(\zeta )\chi _{2}^{2}(\zeta )}}{ (\overline{\chi _{1}(\zeta )\chi _{2}(\zeta )} )^{2}} \leq \frac{1}{4} \biggl(\sqrt{\frac{cd}{CD}}+\sqrt{ \frac{CD}{cd}} \biggr). \end{aligned}$$

(2.10)

Corollary 2.3

Applying Theorem 2.1with $\Delta =(a,b)$, $d\beta (\varsigma )=d\varsigma$,

$$ h(\zeta ,\varsigma )= \textstyle\begin{cases} \frac{g'(\varsigma )}{k\Gamma _{k}(\varrho )(g(\zeta )-g(\varsigma ))^{1-\frac{\varrho }{k}}}, & {a\leq \varsigma \leq \zeta }; \\ 0,& {\zeta < \varsigma \leq b.} \end{cases} $$

(2.11)

Replacing

$$\begin{aligned}& \overline{\bigl(\Upsilon _{1}(\zeta )\Omega _{1}(\zeta )+ \Upsilon _{2}(\zeta )\Omega _{2}(\zeta )\bigr)\chi _{1}(\zeta )\chi _{2}(\zeta )} =I_{a+;g}^{\varrho ,k} \bigl(\Upsilon _{1}(\zeta )\Omega _{1}(\zeta )+ \Upsilon _{2}(\zeta )\Omega _{2}(\zeta )\bigr)\chi _{1}( \zeta )\chi _{2}( \zeta ), \\& \overline{\Omega _{1}(\zeta )\Omega _{2}(\zeta )\chi _{1}^{2}(\zeta )}=I_{a+;g}^{ \varrho ,k}\Omega _{1}(\zeta )\Omega _{2}(\zeta )\chi _{1}^{2}( \zeta ) \end{aligned}$$

and

$$ \overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )\chi _{2}^{2}(\zeta )}=I_{a+;g}^{ \varrho ,k}\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )\chi _{2}^{2}( \zeta ), $$

we get

$$\begin{aligned} &I_{a+;g}^{\varrho ,k}\bigl(\Upsilon _{1}(\zeta )\Omega _{1}(\zeta )+ \Upsilon _{2}(\zeta )\Omega _{2}( \zeta )\bigr)\chi _{1}(\zeta )\chi _{2}( \zeta ) \\ &\quad \geq 4I_{a+;g}^{\varrho ,k}\Omega _{1}(\zeta )\Omega _{2}(\zeta ) \chi _{1}^{2}(\zeta )\cdot I_{a+;g}^{\varrho ,k}\Upsilon _{1}(\zeta ) \Upsilon _{2}(\zeta )\chi _{2}^{2}(\zeta ), \end{aligned}$$

(2.12)

which is [49, Theorem 2.1].

Remark 2.4

In particular, if we choose $k=1$ in Corollary 2.3, we get the following inequality:

$$\begin{aligned} &I_{a+;g}^{\varrho }\bigl(\Upsilon _{1}(\zeta )\Omega _{1}(\zeta )+ \Upsilon _{2}(\zeta )\Omega _{2}( \zeta )\bigr)\chi _{1}(\zeta )\chi _{2}( \zeta ) \\ &\quad \geq 4I_{a+;g}^{\varrho }\Omega _{1}(\zeta )\Omega _{2}(\zeta ) \chi _{1}^{2}(\zeta )\cdot I_{a+;g}^{\varrho }\Upsilon _{1}(\zeta ) \Upsilon _{2}(\zeta )\chi _{2}^{2}(\zeta ), \end{aligned}$$

(2.13)

given in [49, Corollary 3.3].

Remark 2.5

Applying Corollary 2.3 with $\psi (\zeta )=\zeta$, and corresponding $h(\zeta ,\varsigma )$ defined by (2.11) takes the form

$$ h(\zeta ,\varsigma )= \textstyle\begin{cases} \frac{1}{k\Gamma _{k}(\varrho )(\zeta -\varsigma )^{1-\frac{\varrho }{k}}}, & {a\leq \varsigma \leq \zeta }; \\ 0,& {\zeta < \varsigma \leq b,} \end{cases} $$

(2.14)

and (2.13) becomes

$$\begin{aligned} &I_{a+}^{\varrho ,k}\bigl(\Upsilon _{1}(\zeta )\Omega _{1}(\zeta )+ \Upsilon _{2}(\zeta )\Omega _{2}( \zeta )\bigr)\chi _{1}(\zeta )\chi _{2}( \zeta ) \\ &\quad \geq 4I_{a+}^{\varrho ,k}\Omega _{1}(\zeta )\Omega _{2}(\zeta ) \chi _{1}^{2}(\zeta )\cdot I_{a+}^{\varrho ,k}\Upsilon _{1}(\zeta ) \Upsilon _{2}(\zeta )\chi _{2}^{2}(\zeta ), \end{aligned}$$

(2.15)

which leads to [49, Corollary 3.4]. Moreover, if we take $k=1$, then (2.15) becomes the inequality given in [36, Lemma 3.1].

Remark 2.6

Apply Theorem 2.1 with $\Delta =(a,b)$, $d\beta (\varsigma )=d\varsigma$,

$$ h(\zeta ,\varsigma )= \textstyle\begin{cases} \frac{1}{\varsigma k\Gamma _{k}(\varrho )(\log \zeta -\log \varsigma )^{1-\frac{\varrho }{k}}}, & {a\leq \varsigma \leq \zeta }; \\ 0,& {\zeta < \varsigma \leq b.} \end{cases} $$

(2.16)

Replacing

$$\begin{aligned}& \overline{\bigl(\Upsilon _{1}(\zeta )\Omega _{1}(\zeta )+ \Upsilon _{2}(\zeta )\Omega _{2}(\zeta )\bigr)\chi _{1}(\zeta )\chi _{2}(\zeta )} =J_{a_{+}}^{\varrho } \bigl(\Upsilon _{1}(\zeta )\Omega _{1}(\zeta )+ \Upsilon _{2}(\zeta )\Omega _{2}(\zeta )\bigr)\chi _{1}( \zeta )\chi _{2}( \zeta ), \\& \overline{\Omega _{1}(\zeta )\Omega _{2}(\zeta )\chi _{1}^{2}(\zeta )}=J_{a_{+}}^{ \varrho }\Omega _{1}(\zeta )\Omega _{2}(\zeta )\chi _{1}^{2}( \zeta ) \end{aligned}$$

and

$$ \overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )\chi _{2}^{2}(\zeta )}=J_{a_{+}}^{ \varrho }\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )\chi _{2}^{2}( \zeta ), $$

we get the following inequality involving Hadamard fractional integrals:

$$\begin{aligned} &J_{a+}^{\varrho ,k}\bigl(\Upsilon _{1}(\zeta )\Omega _{1}(\zeta )+ \Upsilon _{2}(\zeta )\Omega _{2}( \zeta )\bigr)\chi _{1}(\zeta )\chi _{2}( \zeta ) \\ &\quad \geq 4J_{a+}^{\varrho ,k}\Omega _{1}(\zeta )\Omega _{2}(\zeta ) \chi _{1}^{2}(\zeta )\cdot J_{a+}^{\varrho ,k}\Upsilon _{1}(\zeta ) \Upsilon _{2}(\zeta )\chi _{2}^{2}(\zeta ). \end{aligned}$$

(2.17)

Remark 2.7

Applying Theorem 2.1 with $\Delta =(a,b)$, $d\beta (\varsigma )=d\varsigma$,

$$ h(\zeta ,\varsigma )= \textstyle\begin{cases} \frac{1}{\Gamma (\varrho )} \frac{\sigma \gamma ^{-\sigma (\varrho +\eta )}}{(\gamma ^{\sigma }-\varsigma ^{\sigma })^{1-\varrho }}\varsigma ^{\sigma \eta +\sigma -1},& {a\leq \varsigma \leq \gamma }; \\ 0,& {\gamma < \varsigma \leq b.} \end{cases} $$

(2.18)

If we use the following replacements:

$$\begin{aligned}& \overline{\bigl(\Upsilon _{1}(\zeta )\Omega _{1}(\zeta )+ \Upsilon _{2}(\zeta )\Omega _{2}(\zeta )\bigr)\chi _{1}(\zeta )\chi _{2}(\zeta )} \\& \quad =I_{a_{+};\sigma ;\eta }^{\varrho } \bigl(\Upsilon _{1}(\zeta )\Omega _{1}( \zeta )+\Upsilon _{2}(\zeta )\Omega _{2}(\zeta )\bigr)\chi _{1}( \zeta ) \chi _{2}(\zeta ), \\& \overline{\Omega _{1}(\zeta )\Omega _{2}(\zeta )\chi _{1}^{2}(\zeta )}=I_{a_{+}; \sigma ;\eta }^{\varrho }\Omega _{1}(\zeta )\Omega _{2}(\zeta )\chi _{1}^{2}( \zeta ), \end{aligned}$$

and

$$ \overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )\chi _{2}^{2}(\zeta )}=I_{a_{+}; \sigma ;\eta }^{\varrho }\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ) \chi _{2}^{2}( \zeta ), $$

then we get the following inequality involving Erdélyi–Köber fractional integral:

$$\begin{aligned} &I_{a_{+};\sigma ;\eta }^{\varrho }\bigl(\Upsilon _{1}(\zeta )\Omega _{1}( \zeta )+\Upsilon _{2}(\zeta )\Omega _{2}( \zeta )\bigr)\chi _{1}(\zeta ) \chi _{2}(\zeta ) \\ &\quad \geq 4I_{a_{+};\sigma ;\eta }^{\varrho }\Omega _{1}(\zeta )\Omega _{2}( \zeta )\chi _{2}^{2}(\zeta )\cdot I_{a_{+};\sigma ;\eta }^{\varrho } \Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )\chi _{2}^{2}(\zeta ). \end{aligned}$$

(2.19)

Remark 2.8

Choosing $\beta >0$, $\psi (\zeta )=\frac{\gamma ^{\beta }}{\beta }$, and $k=1$ in Corollary 2.3, we get the inequality for the Katugampola fractional integral operators in the literature [53] and the inequality takes the form

$$\begin{aligned} &{}^{\rho }I_{a_{+}}^{\varrho }\bigl(\Upsilon _{1}( \zeta )\Omega _{1}(\zeta )+ \Upsilon _{2}(\zeta )\Omega _{2}(\zeta )\bigr)\chi _{1}(\zeta )\chi _{2}( \zeta ) \\ &\quad \geq 4{}^{\rho }I_{a_{+}}^{\varrho }\Omega _{1}(\zeta )\Omega _{2}( \zeta )\chi _{1}^{2}( \zeta )\cdot {}^{\rho }I_{a_{+}}^{\varrho }\Upsilon _{1}( \zeta )\Upsilon _{2}(\zeta )\chi _{2}^{2}( \zeta ). \end{aligned}$$

(2.20)

Remark 2.9

Choosing $\beta >0$, $\psi (\zeta )=\frac{(\zeta -a)^{\beta }}{\beta }$, and $k=1$ in Corollary 2.3, we get the following inequality involving conformable fractional integral operators defined by Jarad et al. [51]:

$$\begin{aligned} &{}_{\varrho }^{\beta }\mathfrak{J}^{\varrho }\bigl(\Upsilon _{1}(\zeta ) \Omega _{1}(\zeta )+\Upsilon _{2}( \zeta )\Omega _{2}(\zeta )\bigr)\chi _{1}( \zeta )\chi _{2}(\zeta ) \\ &\quad \geq 4{}_{\varrho }^{\beta }\mathfrak{J}^{\varrho }\Omega _{1}( \zeta )\Omega _{2}(\zeta )\chi _{1}^{2}( \zeta )\cdot {}_{\varrho }^{ \beta }\mathfrak{J}^{\varrho }\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ) \chi _{2}^{2}( \zeta ). \end{aligned}$$

(2.21)

Remark 2.10

Choosing $\beta >0$, $\psi (\zeta )=\frac{\gamma ^{\zeta +\varsigma }}{\zeta +\varsigma }$, and $k=1$ in Corollary 2.3, we get the results involving conformable fractional integral operators defined by Khan et al. [52], i.e.,

$$\begin{aligned} &{}_{\varrho }^{\tau }K_{p^{+}}^{\beta }\bigl(\Upsilon _{1}(\varsigma )\Omega _{1}( \zeta )+\Upsilon _{2}(\zeta )\Omega _{2}(\zeta )\bigr)\chi _{1}( \zeta ) \chi _{2}(\zeta ) \\ &\quad \geq 4{}_{\varrho }^{\tau }K_{p^{+}}^{\beta } \Omega _{1}(\zeta ) \Omega _{2}(\zeta )\chi _{1}^{2}(\zeta )\cdot {}_{\varrho }^{\tau }K_{p^{+}}^{ \beta } \Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )\chi _{2}^{2}(\zeta ). \end{aligned}$$

(2.22)

Lemma 2.11

Let $\chi _{1}$, $\chi _{2}$, $\Upsilon _{1}$, $\Upsilon _{2}$, $\Omega _{1}$, and $\Omega _{2}$ be positive integrable functions defined on $[0,\infty )$, and $\chi _{1}, \chi _{2}, \Upsilon _{1}, \Upsilon _{2}, \Omega _{1}, \Omega _{2}\in \mathfrak{C(h)}$ such that (2.1) holds for all $\varsigma \in [0, \gamma ]$. Then

$$\begin{aligned} \frac{\overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )} \,\overline{\Omega _{1}(\zeta )\Omega _{2}(\zeta )}\, \overline{\chi _{1}^{2}(\zeta )} \,\overline{\chi _{2}^{2}(\zeta )}}{ (\overline{\Upsilon _{1}(\zeta )\chi _{1}(\zeta )} \,\overline{\Omega _{1}(\zeta )\chi _{2}(\zeta )}+\overline{\Upsilon _{2}(\zeta )\chi _{1}(\zeta )} \,\overline{\Omega _{2}(\zeta )\chi _{2}(\zeta )} )^{2}} \leq \frac{1}{4}. \end{aligned}$$

(2.23)

Proof

From (2.1), we have the following fact:

$$\begin{aligned}& \biggl(\frac{\Upsilon _{2}(\varsigma )}{\Omega _{1}(\eta )}- \frac{\chi _{1}(\varsigma )}{\chi _{2}(\eta )} \biggr)\geq 0, \end{aligned}$$

(2.24)

$$\begin{aligned}& \biggl(\frac{\chi _{1}(\varsigma )}{\chi _{2}(\eta )}- \frac{\Upsilon _{1}(\varsigma )}{\Omega _{2}(\eta )} \biggr)\geq 0, \end{aligned}$$

(2.25)

which implies that

$$ \biggl(\frac{\Upsilon _{1}(\varsigma )}{\Omega _{2}(\eta )}+ \frac{\Upsilon _{2}(\varsigma )}{\Omega _{1}(\eta )} \biggr) \frac{\chi _{1}(\varsigma )}{\chi _{2}(\eta )} \geq \frac{\chi _{1}^{2}(\varsigma )}{\chi _{2}^{2}(\eta )}+ \frac{\Upsilon _{1}(\varsigma )\Upsilon _{2}(\varsigma )}{\Omega _{1}(\eta )\Omega _{2}(\eta )}. $$

(2.26)

Multiplying both sides of (2.26) by $\Omega _{1}(\eta )\Omega _{2}(\eta )\chi _{2}^{2}(\eta )$, we obtain

$$\begin{aligned} &\Upsilon _{1}(\varsigma )\chi _{1}(\varsigma )\Omega _{1}(\eta ) \chi _{2}(\eta )+\Upsilon _{2}( \varsigma )\chi _{1}(\varsigma ) \Omega _{2}(\eta )\chi _{2}(\eta ) \\ &\quad \geq \Omega _{1}(\eta )\Omega _{2}(\eta )\chi _{2}^{2}(\varsigma )+ \Upsilon _{1}(\eta )\Upsilon _{2}(\eta )\chi _{2}^{2}(\varsigma ). \end{aligned}$$

(2.27)

Multiplying (2.27) by $h(\gamma ,\zeta )h(\gamma ,\eta )$, then integrating with respect to ζ and η over measure space Δ, we obtain

$$\begin{aligned} &\overline{\Upsilon _{1}(\zeta )\chi _{1}(\zeta )} \,\overline{\Omega _{1}(\zeta )\chi _{2}(\zeta )}+ \overline{ \Upsilon _{2}(\zeta )\chi _{1}(\zeta )} \,\overline{\Omega _{2}(\zeta )\chi _{2}(\zeta )} \\ &\quad \geq \overline{\Omega _{1}(\zeta )\Omega _{2}(\zeta )} \,\overline{\chi _{1}^{2}(\zeta )} + \overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )}\, \overline{\chi _{2}^{2}(\zeta )}. \end{aligned}$$

(2.28)

By arithmetic-geometric mean inequality, we obtain

$$\begin{aligned} &\overline{\Upsilon _{1}(\zeta )\chi _{1}(\zeta )} \,\overline{\Omega _{1}(\zeta )\chi _{2}(\zeta )}+ \overline{ \Upsilon _{2}(\zeta )\chi _{1}(\zeta )} \,\overline{\Omega _{2}(\zeta )\chi _{2}(\zeta )} \\ &\quad \geq 2\sqrt{\overline{\Omega _{1}(\zeta )\Omega _{2}(\zeta )} \,\overline{\chi _{1}^{2}(\zeta )} \,\overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )} \,\overline{\chi _{2}^{2}(\zeta )}}. \end{aligned}$$

(2.29)

This completes the proof. □

Corollary 2.12

Apply Lemma 2.11with $\Delta =(a,b)$, $d\beta (\varsigma )=d\varsigma$, and $h(\gamma ,\varsigma )$ defined by (2.11). Make the following substitutions:

$$\begin{aligned}& \overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )}=I_{a+;g}^{ \varrho ,k}\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ), \\& \overline{\Omega _{1}(\zeta )\Omega _{2}(\zeta )}=I_{a+;g}^{\varrho ,k} \Omega _{1}(\zeta )\Omega _{2}(\zeta ), \\& \overline{\chi _{1}^{2}(\zeta )}=I_{a+;g}^{\varrho ,k} \chi _{1}^{2}( \zeta ), \\& \overline{\chi _{2}^{2}(\zeta )}=I_{a+;g}^{\varrho ,k} \chi _{2}^{2}( \zeta ), \\& \overline{\Upsilon _{1}(\zeta )\chi _{1}(\zeta )}=I_{a+;g}^{\varrho ,k} \Upsilon _{1}(\zeta )\chi _{1}(\zeta ), \\& \overline{\Omega _{2}(\zeta )\chi _{2}(\zeta )}=I_{a+;g}^{\varrho ,k} \Omega _{2}(\zeta )\chi _{2}(\zeta ), \\& \overline{\Upsilon _{2}(\zeta )\chi _{1}(\zeta )}=I_{a+;g}^{\varrho ,k} \Upsilon _{2}(\zeta )\chi _{1}(\zeta ), \end{aligned}$$

and

$$ \overline{\Omega _{1}(\zeta )\chi _{2}(\zeta )}=I_{a+;g}^{\varrho ,k} \Omega _{1}(\zeta )\chi _{2}(\zeta ), $$

we get the following inequality:

$$\begin{aligned} \frac{I_{a+;g}^{\varrho ,k}\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ) I_{a+;g}^{\varrho ,k}\Omega _{1}(\zeta )\Omega _{2}(\zeta ) I_{a+;g}^{\varrho ,k}\chi _{1}^{2}(\zeta ) I_{a+;g}^{\varrho ,k}\chi _{2}^{2}(\zeta )}{ (I_{a+;g}^{\varrho ,k}\Upsilon _{1}(\zeta )\chi _{1}(\zeta ) I_{a+;g}^{\varrho ,k}\Omega _{1}(\zeta )\chi _{2}(\zeta )+I_{a+;g}^{\varrho ,k}\Upsilon _{2}(\zeta ) \chi _{1}(\zeta ) I_{a+;g}^{\varrho ,k}\Omega _{2}(\zeta )\chi _{2}(\zeta ) )^{2}} \leq \frac{1}{4}, \end{aligned}$$

(2.30)

which is [49, Lemma 3.6].

Remark 2.13

In particular, if we choose $k=1$ in Corollary 2.12, we get [49, Corollary 3.1].

Remark 2.14

Applying Corollary 2.3 with $\psi (\zeta )=\zeta$ leads to the inequality in Corollary 3.4 of [49].

Remark 2.15

Apply Theorem 2.1 with $\Delta =(a,b)$, $d\beta (\varsigma )=d\varsigma$, defined by (2.16). Replacing

$$\begin{aligned}& \overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )}=J_{a_{+}}^{ \varrho }\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ), \\& \overline{\Omega _{2}(\zeta )\Omega _{2}(\zeta )}=J_{a_{+}}^{\varrho } \Omega _{2}(\zeta )\Omega _{2}(\zeta ), \\& \overline{\chi _{1}^{2}(\zeta )}=J_{a_{+}}^{\varrho } \chi _{1}^{2}( \zeta ), \\& \overline{\chi _{2}^{2}(\zeta )}=J_{a_{+}}^{\varrho } \chi _{2}^{2}( \zeta ), \\& \overline{\Omega _{2}(\zeta )\chi _{2}(\zeta )}=J_{a_{+}}^{\varrho } \Omega _{2}(\zeta )\chi _{2}(\zeta ), \\& \overline{\Upsilon _{2}(\zeta )\chi _{1}(\zeta )}=J_{a_{+}}^{\varrho } \Upsilon _{2}(\zeta )\chi _{1}(\zeta ), \\& \overline{\Omega _{1}(\zeta )\chi _{2}(\zeta )}=J_{a_{+}}^{\varrho } \Omega _{1}(\zeta )\chi _{2}(\zeta ), \end{aligned}$$

and

$$ \overline{\Upsilon _{1}(\zeta )\chi _{1}(\zeta )}=J_{a_{+}}^{\varrho } \Upsilon _{1}(\zeta )\chi _{1}(\zeta ), $$

we get the inequality for Hadamard fractional integral, i.e.,

$$\begin{aligned} \frac{J_{a_{+}}^{\varrho }\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ) J_{a_{+}}^{\varrho }\Omega _{1}(\zeta )\Omega _{2}(\zeta ) J_{a_{+}}^{\varrho }\chi _{1}^{2} (\zeta ) J_{a_{+}}^{\varrho }\chi _{2}^{2}(\zeta )}{ (J_{a_{+}}^{\varrho }\Upsilon _{1}(\zeta )\chi _{1}(\zeta ) J_{a_{+}}^{\varrho }\Omega _{1}(\zeta )\chi _{2}(\zeta )+J_{a_{+}}^{\varrho } \Upsilon _{2}(\zeta )\chi _{1}(\zeta ) J_{a_{+}}^{\varrho }\Omega _{2}(\zeta )\chi _{2}(\zeta ) )^{2}} \leq \frac{1}{4}. \end{aligned}$$

(2.31)

Remark 2.16

Apply Theorem 2.1 with $\Delta =(a,b)$, $d\beta (\varsigma )=d\varsigma$, $h(\zeta ,\varsigma )$ defined by (2.18). Replacing

$$\begin{aligned}& \overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )}=I_{a_{+}; \sigma ;\eta }^{\varrho }\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ), \\& \overline{\Upsilon _{2}(\varsigma )\Upsilon _{2}(\zeta )}=I_{a_{+}; \sigma ;\eta }^{\varrho }\Upsilon _{2}(\zeta )\Upsilon _{2}(\zeta ), \\& \overline{\chi _{2}^{2}(\zeta )}=I_{a_{+};\sigma ;\eta }^{\varrho } \chi _{1}^{2}(\zeta ), \\& \overline{\chi _{2}^{2}(\zeta )}=I_{a_{+};\sigma ;\eta }^{\varrho } \chi _{2}^{2}(\zeta ), \\& \overline{\Omega _{2}(\zeta )\chi _{2}(\zeta )}=I_{a_{+};\sigma ;\eta }^{ \varrho }\Omega _{2}(\zeta )\chi _{2}(\zeta ), \\& \overline{\Upsilon _{2}(\zeta )\chi _{1}(\zeta )}=I_{a_{+};\sigma ; \eta }^{\varrho }\Upsilon _{2}(\zeta )\chi _{1}(\zeta ), \\& \overline{\Omega _{1}(\zeta )\chi _{2}(\zeta )}=I_{a_{+};\sigma ;\eta }^{ \varrho }\Omega _{1}(\zeta )\chi _{2}(\zeta ), \end{aligned}$$

and

$$ \overline{\Upsilon _{1}(\zeta )\chi _{1}(\zeta )}=I_{a_{+};\sigma ; \eta }^{\varrho }\Upsilon _{1}(\zeta )\chi _{1}(\zeta ), $$

we get the inequality for Erdélyi–Köber type fractional integral, i.e.,

$$\begin{aligned} \frac{I_{a_{+};\sigma ;\eta }^{\varrho }\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ) I_{a_{+};\sigma ;\eta }^{\varrho }\Omega _{1} (\zeta )\Omega _{2}(\zeta ) I_{a_{+};\sigma ;\eta }^{\varrho }G^{2}(\zeta ) I_{a_{+};\sigma ;\eta }^{\varrho }\chi _{2}^{2}(\zeta )}{ (I_{a_{+};\sigma ;\eta }^{\varrho }\Upsilon _{1}(\zeta )\chi _{1}(\zeta ) I_{a_{+};\sigma ;\eta }^{\varrho }\Omega _{1} (\zeta )\chi _{2}(\zeta )+I_{a_{+};\sigma ;\eta }^{\varrho }\Upsilon _{2}(\zeta )\chi _{1}(\zeta ) I_{a_{+};\sigma ;\eta }^{\varrho }\Omega _{2}(\zeta )\chi _{2}(\zeta ) )^{2}} \leq \frac{1}{4}. \end{aligned}$$

Remark 2.17

Choosing $\beta >0$, $\psi (\zeta )=\frac{\zeta ^{\beta }}{\beta }$, and $k=1$ in Corollary 2.12, we get the inequality for the Katugampola fractional integral operator, and the inequality takes the form after replacing

$$\begin{aligned}& \overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )}= {{}^{\rho }I_{a_{+}}^{ \varrho }}\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ), \\& \overline{\Upsilon _{2}(\zeta )\Upsilon _{2}(\zeta )}={{}^{\rho }I_{a_{+}}^{ \varrho }}\Upsilon _{2}( \zeta )\Upsilon _{2}(\zeta ), \\& \overline{\chi _{1}^{2}(\zeta )}={{}^{\rho }I_{a_{+}}^{\varrho }} \chi _{1}^{2}( \zeta ), \\& \overline{\chi _{2}^{2}(\zeta )}={{}^{\rho }I_{a_{+}}^{\varrho }} \chi _{2}^{2}( \zeta ), \\& \overline{\Omega _{2}(\zeta )\chi _{2}(\zeta )}={{}^{\rho }I_{a_{+}}^{ \varrho }}\Omega _{2}(\zeta )H(\zeta ), \\& \overline{\Upsilon _{2}(\zeta )\chi _{1}(\zeta )}={{}^{\rho }I_{a_{+}}^{ \varrho }}\Upsilon _{2}( \zeta )\chi _{1}(\zeta ), \\& \overline{\Omega _{1}(\zeta )\chi _{2}(\zeta )}={{}^{\rho }I_{a_{+}}^{ \varrho }}\Omega _{1}(\zeta )\chi _{2}(\zeta ), \end{aligned}$$

and

$$ \overline{\Upsilon _{1}(\zeta )\chi _{1}(\zeta )}={{}^{\rho }I_{a_{+}}^{ \varrho }}\Upsilon _{1}( \zeta )\chi _{1}(\zeta ), $$

we get

$$\begin{aligned} \frac{{{}^{\rho }I_{a_{+}}^{\varrho }}\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ) {{}^{\rho }I_{a_{+}}^{\varrho }}\Omega _{1}(\zeta ) \Omega _{2}(\zeta ) {{}^{\rho }I_{a_{+}}^{\varrho }}\chi _{1}^{2}(\zeta ) {{}^{\rho }I_{a_{+}}^{\varrho }}\chi _{2}^{2}(\zeta )}{ ({{}^{\rho }I_{a_{+}}^{\varrho }}\Upsilon _{1}(\zeta )\chi _{1}(\zeta ) {{}^{\rho }I_{a_{+}}^{\varrho }} \Omega _{1}(\zeta )\chi _{2}(\zeta )+{{}^{\rho }I_{a_{+}}^{\varrho }}\Upsilon _{2}(\zeta )\chi _{1}(\zeta ) {{}^{\rho }I_{a_{+}}^{\varrho }}\Omega _{2}(\zeta )\chi _{2}(\zeta ) )^{2}} \leq \frac{1}{4}. \end{aligned}$$

(2.32)

Remark 2.18

Choosing $\beta >0$, $\psi (\zeta )=\frac{(\zeta -a)^{\beta }}{\beta }$, and $k=1$ in Corollary 2.12, we get the inequality involving conformable fractional integral operators defined by Jarad et al. [51]. Replacing

$$\begin{aligned}& \overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )}={{}_{\varrho }^{ \rho }\mathfrak{J}^{\varrho }}\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ), \\& \overline{\Upsilon _{2}(\zeta )\Upsilon _{2}(\zeta )}={{}_{\varrho }^{ \rho }\mathfrak{J}^{\varrho }}\Upsilon _{2}(\zeta )\Upsilon _{2}(\zeta ), \\& \overline{\chi _{1}^{2}(\zeta )}={{}_{\varrho }^{\rho } \mathfrak{J}^{ \varrho }}\chi _{1}^{2}(\zeta ), \\& \overline{\chi _{2}^{2}(\zeta )}={{}_{\varrho }^{\rho } \mathfrak{J}^{ \varrho }}\chi _{2}^{2}(\zeta ), \\& \overline{\Omega _{2}(\zeta )\chi _{2}(\zeta )}={{}_{\varrho }^{\rho } \mathfrak{J}^{\varrho }}\Omega _{2}(\zeta )\chi _{2}(\zeta ), \\& \overline{\Upsilon _{2}(\zeta )\chi _{1}(\zeta )}={{}_{\varrho }^{\rho } \mathfrak{J}^{\varrho }}\Upsilon _{2}(\zeta )\chi _{1}(\zeta ), \\& \overline{\Omega _{1}(\zeta )\chi _{2}(\zeta )}={{}_{\varrho }^{\rho } \mathfrak{J}^{\varrho }}\Omega _{1}(\zeta )\chi _{2}(\zeta ), \end{aligned}$$

and

$$ \overline{\Upsilon _{1}(\zeta )\chi _{1}(\zeta )}={{}_{\varrho }^{\rho } \mathfrak{J}^{\varrho }}\Upsilon _{1}(\zeta )\chi _{1}(\zeta ), $$

we get

$$\begin{aligned} \frac{{{}_{\varrho }^{\rho }\mathfrak{J}^{\varrho }}\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ) {{}_{\varrho }^{\rho } \mathfrak{J}^{\varrho }}\Omega _{1}(\zeta )\Omega _{2}(\zeta ) {{}_{\varrho }^{\rho }\mathfrak{J}^{\varrho }}\chi _{1}^{2}(\zeta ) {{}_{\varrho }^{\rho }\mathfrak{J}^{\varrho }}\chi _{2}^{2}(\zeta )}{ ({{}_{\varrho }^{\rho }\mathfrak{J}^{\varrho }}\Upsilon _{1}(\zeta )\chi _{1}(\zeta ) {{}_{\varrho }^{\rho }\mathfrak{J}^{\varrho }}\Omega _{1}(\zeta )\chi _{2}(\zeta ) +{{}_{\varrho }^{\rho }\mathfrak{J}^{\varrho }}\Upsilon _{2}(\zeta )\chi _{1}(\zeta ) {{}_{\varrho }^{\rho }\mathfrak{J}^{\varrho }}\Omega _{2}(\zeta )\chi _{2}(\zeta ) )^{2}} \leq \frac{1}{4}. \end{aligned}$$

(2.33)

Remark 2.19

Choosing $\beta >0$, $\psi (\zeta )=\frac{\zeta ^{\xi +\varsigma }}{\xi +\varsigma }$, and $k=1$ in Corollary 2.12, we get the inequality for the conformable fractional integral operators defined by Khan et al. [52], and the inequality takes the form by replacing

$$\begin{aligned}& \overline{\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta )}={{}_{\varrho }^{ \tau }K_{p^{+}}^{\beta }}\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ), \\& \overline{\Upsilon _{2}(\zeta )\Upsilon _{2}(\zeta )}={{}_{\varrho }^{ \tau }K_{p^{+}}^{\beta }}\Upsilon _{2}(\zeta )\Upsilon _{2}(\zeta ), \\& \overline{\chi _{1}^{2}(\zeta )}={{}_{\varrho }^{\tau }K_{p^{+}}^{\beta }}G^{2}( \zeta ), \\& \overline{\chi _{2}^{2}(\zeta )}={{}_{\varrho }^{\tau }K_{p^{+}}^{\beta }} \chi _{2}^{2}(\zeta ), \\& \overline{\Omega _{2}(\zeta )\chi _{2}(\zeta )}={{}_{\varrho }^{\tau }K_{p^{+}}^{ \beta }}\Omega _{2}(\zeta )\chi _{2}(\zeta ), \\& \overline{\Upsilon _{2}(\zeta )\chi _{1}(\zeta )}={{}_{\varrho }^{\tau }K_{p^{+}}^{ \beta }}\Upsilon _{2}(\zeta )\chi _{1}(\zeta ), \\& \overline{\Omega _{1}(\zeta )\chi _{2}(\zeta )}={{}_{\varrho }^{\tau }K_{p^{+}}^{ \beta }}\Omega _{1}(\zeta )\chi _{2}(\zeta ), \end{aligned}$$

and

$$ \overline{\Upsilon _{1}(\zeta )\chi _{1}(\zeta )}={{}_{\varrho }^{\tau }K_{p^{+}}^{ \beta }}\Upsilon _{1}(\zeta )\chi _{1}(\zeta ), $$

we get

$$\begin{aligned} \frac{{{}_{\varrho }^{\tau }K_{p^{+}}^{\beta }}\Upsilon _{1}(\zeta )\Upsilon _{2}(\zeta ) {{}_{\varrho }^{\tau }K_{p^{+}}^{\beta }}\Omega _{1}(\zeta )\Omega _{2}(\zeta ) {{}_{\varrho }^{\tau }K_{p^{+}}^{\beta }}\chi _{1}^{2}(\zeta ) {{}_{\varrho }^{\tau }K_{p^{+}}^{\beta }}\chi _{2}^{2}(\zeta )}{ ({{}_{\varrho }^{\tau }K_{p^{+}}^{\beta }}\Upsilon _{1}(\zeta )\chi _{1}(\zeta ) {{}_{\varrho }^{\tau }K_{p^{+}}^{\beta }}\Omega _{1}(\zeta )\chi _{2}(\zeta )+ {{}_{\varrho }^{\tau }K_{p^{+}}^{\beta }}\Upsilon _{2}(\zeta )\chi _{1}(\zeta ) {{}_{\varrho }^{\tau }K_{p^{+}}^{\beta }}\Omega _{2}(\zeta )\chi _{2}(\zeta ) )^{2}} \leq \frac{1}{4}. \end{aligned}$$

(2.34)

Theorem 2.20

Let $\chi _{1}, \chi _{2}, \Upsilon _{1}, \Upsilon _{2}, \Omega _{1}, \Omega _{2}\in \mathfrak{C(h)}$ be positive integrable functions defined on $[0,\infty )$ such that (2.1) holds for all $\varsigma \in [0, \zeta ]$. Then we have

$$ \overline{ \biggl(\frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta )} \,\overline{ \biggl(\frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta )} \geq \overline{\chi _{1}^{2}(\zeta )} \,\overline{\chi _{2}^{2}( \zeta )}. $$

(2.35)

Proof

Clearly from (2.1) we can write

$$ \frac{\Upsilon _{2}(\varsigma )}{\Omega _{1}(\varsigma )}\chi _{1}( \varsigma )\chi _{2}(\varsigma )\geq \chi _{1}^{2}(\varsigma ), $$

which implies that

$$ \int _{\Delta }h(\zeta ,\varsigma ) \frac{\Upsilon _{2}(\varsigma )}{\Omega _{1}(\varsigma )}\chi _{1}( \varsigma )\chi _{2}(\varsigma )\,d\beta (\varsigma ) \geq \int _{ \Delta }h(\zeta ,\varsigma ) \chi _{1}^{2}( \varsigma )\,d\beta ( \varsigma ), $$

and (2.35) follows. □

Corollary 2.21

Apply Theorem 2.20with $\Delta =(a,b)$, $d\beta (\varsigma )=d\varsigma$, and $h(\zeta ,\varsigma )$ defined by (2.11). Replacing

$$\begin{aligned}& \overline{ \biggl(\frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta )}=I_{a+;g}^{ \varrho ,k} \biggl( \frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta ), \\& \overline{ \biggl(\frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta )}=I_{a+;g}^{ \varrho ,k} \biggl( \frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta ), \\& \overline{\chi _{1}^{2}(\zeta )} =I_{a+;g}^{\varrho ,k} \chi _{1}^{2}( \zeta ), \end{aligned}$$

and

$$ \overline{\chi _{2}^{2}(\zeta )}=I_{a+;g}^{\varrho ,k} \chi _{2}^{2}( \zeta ), $$

we get

$$\begin{aligned} I_{a+;g}^{\varrho ,k} \biggl( \frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta ) I_{a+;g}^{\varrho ,k} \biggl( \frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta ) \geq I_{a+;g}^{\varrho ,k}\chi _{1}^{2}(\zeta ) I_{a+;g}^{\varrho ,k} \chi _{2}^{2}(\zeta ), \end{aligned}$$

(2.36)

which is [49, Theorem 3.11].

Remark 2.22

In particular, if we choose $k=1$ in Corollary 2.21, we get the following inequality:

$$\begin{aligned} I_{a+;g}^{\varrho } \biggl( \frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta ) I_{a+;g}^{\varrho } \biggl( \frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta ) \geq I_{a+;g}^{\varrho }\chi _{1}^{2}(\zeta ) I_{a+;g}^{\varrho } \chi _{2}^{2}(\zeta ), \end{aligned}$$

(2.37)

given in [49, Corollary 3.13].

Remark 2.23

Applying Corollary 2.21 with $\psi (\zeta )=\zeta$, and corresponding $h(\zeta ,\varsigma )$ takes the form given in (2.14) and the inequality takes the form

$$\begin{aligned} I_{a+;g}^{\varrho ,k} \biggl( \frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta ) I_{a+;g}^{\varrho ,k} \biggl( \frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta ) \geq I_{a+;g}^{\varrho ,k}\chi _{1}^{2}( \zeta ) I_{a+;g}^{\varrho ,k} \chi _{2}^{2}(\zeta ), \end{aligned}$$

(2.38)

which leads to [49, Corollary 3.14]. Moreover, if we take $k=1$, then (2.38) becomes the inequality given in [36, Lemma 3.1].

Remark 2.24

Apply Theorem 2.20 with $\Delta =(a,b)$, $d\beta (\varsigma )=d\varsigma$, and $h(\zeta ,\varsigma )$ defined by (2.16). Replace

$$\begin{aligned}& \overline{ \biggl(\frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr)} =J_{a_{+}}^{\varrho } \biggl( \frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr), \\& \overline{ \biggl(\frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr)}=J_{a_{+}}^{ \varrho } \biggl(\frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr), \\& \overline{\chi _{1}^{2}(\zeta )}=J_{a_{+}}^{\varrho } \chi _{1}^{2}( \zeta ), \end{aligned}$$

and

$$ \overline{\chi _{2}^{2}(\zeta )}=J_{a_{+}}^{\varrho } \chi _{2}^{2}( \zeta ), $$

to get the inequality for the Hadamard type fractional integrals, i.e.,

$$\begin{aligned} J_{a_{+}}^{\varrho } \biggl( \frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta ) J_{a_{+}}^{\varrho } \biggl( \frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta ) \geq J_{a_{+}}^{\varrho }\chi _{1}^{2}(\zeta ) J_{a_{+}}^{\varrho } \chi _{2}^{2}(\zeta ). \end{aligned}$$

(2.39)

Remark 2.25

Apply Theorem 2.21 with $\Delta =(a,b)$, $d\beta (\varsigma )=d\varsigma$, and $h(\zeta ,\varsigma )$ defined by (2.18). Replacing

$$\begin{aligned}& \overline{ \biggl(\frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta )} =I_{a_{+};\sigma ;\eta }^{\varrho } \biggl( \frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta ), \\& \overline{ \biggl(\frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta )} =I_{a_{+};\sigma ;\eta }^{\varrho } \biggl( \frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta ), \\& \overline{\chi _{1}^{2}(\zeta )} =I_{a_{+};\sigma ;\eta }^{\varrho } \chi _{1}^{2}(\zeta ), \end{aligned}$$

and

$$ \overline{\chi _{2}^{2}(\zeta )} =I_{a_{+};\sigma ;\eta }^{\varrho } \chi _{2}^{2}(\zeta ), $$

we get the results involving Erdélyi–Köber fractional integral, i.e.,

$$\begin{aligned} I_{a_{+};\sigma ;\eta }^{\varrho } \biggl( \frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta ) I_{a_{+};\sigma ;\eta }^{\varrho } \biggl( \frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta ) \geq I_{a_{+};\sigma ;\eta }^{\varrho }\chi _{1}^{2}(\zeta ) I_{a_{+}; \sigma ;\eta }^{\varrho }\chi _{2}^{2}(\zeta ). \end{aligned}$$

(2.40)

Remark 2.26

Choosing $\beta >0$, $\psi (\zeta )=\frac{\zeta ^{\beta }}{\beta }$, and $k=1$ in Corollary 2.21, we get the inequality for the Katugampola fractional integral operators [53], and the inequality takes the form

$$\begin{aligned} {}^{\rho }I_{a_{+}}^{\varrho } \biggl( \frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta ) {}^{\rho }I_{a_{+}}^{\varrho } \biggl( \frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta ) \geq {}^{\rho }I_{a_{+}}^{\varrho } \chi _{1}^{2}(\zeta ){}^{\rho }I_{a_{+}}^{ \varrho } \chi _{2}^{2}(\zeta ). \end{aligned}$$

(2.41)

Remark 2.27

Choosing $\beta >0$, $\psi (\zeta )=\frac{(\zeta -a)^{\beta }}{\beta }$, and $k=1$ in Corollary 2.21, we get the inequality for the conformable fractional integral operators defined by Jarad et al. [51], and the inequality takes the form:

$$\begin{aligned} {}_{\varrho }^{\rho }\mathfrak{J}^{\varrho } \biggl( \frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta ) {}_{\varrho }^{\rho } \mathfrak{J}^{\varrho } \biggl( \frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta ) \geq {{}_{\varrho }^{\rho }\mathfrak{J}^{\varrho }}\chi _{1}^{2}(\zeta ) {}_{\varrho }^{\rho } \mathfrak{J}^{\varrho }\chi _{2}^{2}(\zeta ). \end{aligned}$$

(2.42)

Remark 2.28

Choosing $\beta >0$, $\psi (\zeta )=\frac{(\zeta )^{\xi +\varsigma }}{\xi +\varsigma }$, and $k=1$ in Corollary 2.21, we get the inequality for the conformable fractional integral operators defined by Khan et al. [52], and the inequality takes the form

$$\begin{aligned} {}_{\varrho }^{\tau }K_{p^{+}}^{\beta } \biggl( \frac{\Upsilon _{2} \chi _{1}\chi _{2}}{\Omega _{1}} \biggr) (\zeta ) {}_{\varrho }^{\tau }K_{p^{+}}^{\beta } \biggl( \frac{\Omega _{2} \chi _{1}\chi _{2}}{\Upsilon _{1}} \biggr) (\zeta ) \geq {}_{\varrho }^{\tau }K_{p^{+}}^{\beta } \chi _{1}^{2}(\zeta ){}_{ \varrho }^{\tau }K_{p^{+}}^{\beta } \chi _{2}^{2}(\zeta ). \end{aligned}$$

(2.43)

3 Čebyšev type inequalities

In this section, we shall present several Čebyšev type inequalities involving general kernels and application in fractional integrals.

Theorem 3.1

Let $\chi _{1}$ and $\chi _{2}$ be two integrable and synchronous functions on $[0,\infty )$ and $\chi _{1}, \chi _{2}\in \mathfrak{C(h)}$. Then one has

$$ \overline{(\chi _{1}\chi _{2}) (\zeta )}\geq \frac{1}{\Upsilon (\zeta )} \,\overline{\chi _{1}(\zeta )} \,\overline{\chi _{2}(\zeta )}, $$

(3.1)

where ϒ is defined by (1.3).

Proof

By using the synchronism property for the functions $\chi _{1}$ and $\chi _{2}$ on the interval $[0,\infty )$, we get

$$ \chi _{1}(\zeta )\chi _{2}(\zeta ) + \chi _{1}( \varsigma )\chi _{2}( \varsigma ) \geq \chi _{1}(\zeta )\chi _{2}(\varsigma ) + \chi _{1}( \varsigma )\chi _{2}(\zeta ). $$

Multiplying by $h(\zeta ,\varsigma )$ and integrating with respect to ς over measure space Δ leads to

$$\begin{aligned} & \int _{\Delta } h(\zeta ,\varsigma )\chi _{1}(\varsigma ) \chi _{2}(\varsigma ) \,d\beta (\varsigma ) + \int _{\Delta }h( \zeta ,\varsigma )\chi _{1}(\xi )\chi _{2}(\xi )\,d\beta (\varsigma ) \\ &\quad \geq \int _{\Delta }h(\zeta ,\varsigma ) \chi _{1}(\zeta ) \chi _{2}(\xi )\,d\beta (\varsigma ) + \int _{\Delta }h(\zeta , \varsigma ) \chi _{1}(\xi )\chi _{2}(\varsigma ) \,d\beta (\varsigma ). \end{aligned}$$

(3.2)

Therefore, we get that

$$ \overline{ (\chi _{1}\chi _{2}) (\zeta )}+ \chi _{1}(\xi ) \chi _{2}( \xi )\Upsilon (\zeta )\geq \chi _{2}(\xi ) \,\overline{\chi _{1}(\zeta )}+\chi _{1}(\xi ) \,\overline{\chi _{2}(\zeta )}. $$

(3.3)

Multiplying by $h(\zeta ,\xi )$ and integrating with respect to ξ over Δ leads to

$$ \Upsilon (\zeta )\,\overline{(\chi _{1}\chi _{2}) (\zeta )}+ \overline{\chi _{1}(\zeta )\chi _{2}(\zeta )}\Upsilon (\zeta )\geq \overline{\chi _{2}(\zeta )\chi _{1}(\zeta )}+ \overline{\chi _{2}(\zeta )}\,\overline{\chi _{1}(\zeta )}. $$

That implies (3.1). □

Corollary 3.2

Applying Theorem 3.1with $\Delta =(a,b)$, $d\beta (\varsigma )=d\varsigma$, and $h(\zeta ,\varsigma )$ defined by (2.11), after some calculations, we get

$$ \Upsilon (\zeta )=\frac{1}{\Gamma _{k}(\varrho +k)}\bigl(\psi (\zeta )- \psi (a) \bigr)^{\frac{\varrho }{k}}. $$

Replacing $\overline{(\chi _{1}\chi _{2})(\zeta )}=I_{a+;\psi }^{\varrho ,k}( \chi _{1}\chi _{2})(\zeta )$, $\overline{\chi _{1}(\zeta )}=I_{a+;\psi }^{\varrho ,k}\chi _{1}( \zeta )$, and $\overline{\chi _{2}(\zeta )}=I_{a+;\psi }^{\varrho ,k}\chi _{2}( \zeta )$, we get

$$ I_{a+;\psi }^{\varrho ,k}(\chi _{1}\chi _{2}) (\zeta )\geq \frac{1}{\Upsilon (\zeta )} I_{a+;\psi }^{\varrho ,k} \chi _{1}(\zeta ) \cdot I_{a+;\psi }^{\varrho ,k}\chi _{2}(\zeta ), $$

(3.4)

which is [49, Theorem 4.1].

Remark 3.3

In particular, if we choose $k=1$ in Corollary 3.2, we get [49, Corollary 4.2].

Remark 3.4

In particular, if we choose $\psi (\zeta )=\zeta$ in Corollary 3.2, we get [49, Corollary 4.3].

Remark 3.5

In particular, if we choose $\psi (\zeta )=\zeta$, $k=1$ in Corollary 3.2, we get [49, Corollary 4.4].

Remark 3.6

In particular, if we choose $\psi (\zeta )=\log (\zeta )$, $k=1$ in Corollary 3.2, it leads to the inequality for Katugampola fractional integral operator [53].

Remark 3.7

If we choose $\beta >0$, $\psi (\zeta )=\frac{(\zeta -a)^{\beta }}{\beta }$, and $k=1$ in Corollary 3.2, we get the inequality for the conformable fractional integral operator [51].

Remark 3.8

Choosing $\beta >0$, $\psi (\zeta )=\frac{\zeta ^{\xi +\varsigma }}{\xi +\varsigma }$, and $k=1$ in Corollary 3.2, we get the inequality for the conformable fractional integral operator [52].

Theorem 3.9

Let $\chi _{j}$, ($1\leq j\leq \gamma$) be integrable and synchronous functions on $[0,\infty )$ and $\chi _{j} \in \mathfrak{C(h)}$. Then

$$ \overline{\prod_{j=1}^{\gamma }\chi _{j}}(\zeta )\leq \biggl( \frac{1}{\Upsilon (\zeta )} \biggr)\prod _{j=1}^{\gamma } \,\overline{\chi _{j}}. $$

Proof

Proof follows from the mathematical induction. □

Remark 3.10

Like previous remarks we can give the applications for fractional integrals discussed in the paper. But we omit the details here.

4 Concluding remarks

In the ongoing years numerous analysts have given the generalization of integral operators and constructed fruitful inequalities. For us it is always interesting and motivating to give the generalization of all previous results. Inspired by the above-mentioned results, we introduced certain rich inequalities successfully, which generalized all the previous results. We developed a class of functions representing the integral transform with general kernel. We proved a wide range of Pólya–Szegö and Čebyšev type inequalities involving general kernel over σ-finite measure. We extracted the known results from our general results.