Abstract
The aim of this paper is first to introduce generalizations of quantum integrals and derivatives which are called \((\phi \,-\,h)\) integrals and \((\phi \,-\,h)\) derivatives, respectively. Then we investigate some implicit integral inequalities for \((\phi \,-\,h)\) integrals. Different classes of convex functions are used to prove these inequalities for symmetric functions. Under certain assumptions, Hermite–Hadamard-type inequalities for q-integrals are deduced. The results presented herein are applicable to convex, m-convex, and \(\hbar \)-convex functions defined on the non-negative part of the real line.
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1 Introduction
Theory of integral inequalities provides useful tools to estimate an integral of functions over intervals. These inequalities give bounds to estimate the cumulative behavior of functions over a specified domain and provide insight into the general behavior of different mathematical objects. On the other hand, Quantum calculus known as q-Calculus is based on the idea of studying the concepts of classical calculus without taking into accounts the limits. In the nineteenth century, Euler [14] investigated several notable results such as Jacobi’s triple product identity and theory of q-hypergeometric functions. The study of quantum integral inequalities has gained much attention of researchers these days. The concept of the definite q-integral was first introduced by Jackson [13] in 1910. Indeed, he was the first who developed q-calculus in a systematic way. In, 2013 and 2014, Tariboon and Ntouyas [23, 24] introduced the concepts of q-derivatives and q-integrals over finite intervals and derived quantum integral inequalities. In 2018, Alp et al. [3] employed the left q-integral operator to develop q-Hermite–Hadamard inequality using convex and quasi-convex functions. Also, Alomari [2] derived the quantum variant of Bernoulli inequality. In 2022, Liu et al. [15] proved the q-Hermite–Hadamard inequality by using \((q-h)\) -derivatives and integrals. Recently, Chen et al. [7] derived some results for \((q-h)\)-integrals using \(\hbar \)-convex and m-convex functions and Farid et al. [9] proved the generalized q-integral inequalities using \((\hbar -m)\) and \((\alpha -m)\) convexities. For some other recent developments in this direction, we refer to [4,5,6, 8, 10, 11, 14, 16, 17, 22] and references mentioned therein.In this paper, a new version of Hermite–Hadamard quantum integral inequalities for convex, m-convex and \(\hbar \)-convex functions using the left and right \((\phi -h)\)-integrals are obtained. The results of this paper can be applied to convex functions defined on non negative part of real line \(\mathbb {R}\).
Definition 1
[20] If I is any interval in \(\mathbb {R}\). A function \(f:{I} \subseteq \mathbb {R}\rightarrow \mathbb {R}\) is said to be convex function if
If the inequality is reversed, then f is concave function.
Definition 2
[26] Suppose that \({I}\text { and } {\Omega } \) are any two intervals in \( \mathbb {R}\) and \(\hbar :(0, 1)\subseteq {\Omega }\rightarrow \mathbb {R}\) is a nonnegative function. A nonnegative function \(f:{I}\subseteq \mathbb {R} \rightarrow \mathbb {R} \) is said to be \({\hbar }\)-convex if
If the inequality is reversed then f is \({\hbar }\)-concave. If \({ \hbar (\lambda )=\lambda }\) in (1.2), then the above inequality reduces to (1.1), and if we take, \({\hbar (\lambda )=\lambda ^{s}}\) in (1.2), we get the definition of s-convex function given in [12].
Definition 3
[25] Let \(c>0\) and \(0\le m\le 1\). The function \(f:[0,c]\rightarrow \mathbb {R}\) is called m-convex, if the following inequality holds
If \(m=1\) in (1.3), then we get (1.1). If we take, \(m=0\), then we get a star shaped function. That is, for any point \(x\in [0,c]\text { and}\,t\in [0,1]\) we have \(f(tx)\le tf(x)\).
Theorem 1
[19] If \(f: [a, b] \rightarrow \mathbb {R}\) is a convex function, where \( a<b\) and \([a, b]\subset \mathbb {R}\), then, we have the Hermite–Hadamard inequality which is given as follows:
Definition 4
[20] Let f be a convex function defined on an open interval (a, b), \(x_{i}\in (a, b)\) and \(\alpha _{i}>0\) with \(\sum _{i=1}^{n}\alpha _{i}=1\). Then, the following inequality
is called the Jensen inequality and the function f is known as a J -convex function. The following result is an extension of Hermite–Hadamard inequality to the \( \hbar \)-convex function [21].
Theorem 2
If \(f:I \rightarrow \mathbb {R}\) is a \(\hbar \)-convex function on the interval \( I\subset \mathbb {R}\) and \(a, b \in {I}\) where \(a<b\). Then, the generalized form of Hermite–Hadamard inequality is given as follows:
In an ordinary calculus, the classical derivative of a function f is defined as follows:
Quantum derivatives without involving limit have been defined in various ways. We present two types of derivatives known as q-derivative and the h -derivative for a given function f.Let us recall the notations of a quantum differential given as.
Definition 5
[14] The q-differential is:
In particular, \(d_{q}x=(q-1)x\) and \(d_{h}x=h\). These two quantum differentials allow us to define the corresponding quantum derivatives.
Definition 6
[14] For a continuous function \(f:I\rightarrow \mathbb {R}\),
are called the q-derivative and h-derivative, respectively. Note that: If f is differentiable, then we have \(\lim \limits _{q\rightarrow 1}D_{q}f(x)=\lim \limits _{h\rightarrow 0}D_{h}f(x)=\frac{df(x)}{dx}\).
Definition 7
[10] For a continuous function \(f:I\rightarrow \mathbb {R}\), the \((q-h)\)-derivative is expressed as:
where \(q\in (0, 1)\) and \(h\in \mathbb {R }.\) If \(x=x_{0}\), then we have \( D_{q}^{h}f(x_{0})=\lim \limits _{x\rightarrow x_{0}}D_{q}^{h}f(x)\).
When \(h=0\) and \(q=1\) in (7), we get quantum derivative and plank‘s derivative, respectively defined in (6).The Jackson integral or q-integral according to [13], is expressed as:
Definition 8
For a continuous function f, \(0<q<1\) the q-integral on [0, a] is expressed as:
Note that,
If \(c\in (0, b)\), then q-integral of f is defined as:
Definition 9
[24] Assume that \(f:[a,b]\rightarrow \mathbb {R}\) is a continuous function, \(x\in [a,b]\), and \(0<q<1\). Then \(q_{a}\)-definite integral on [a, b] is defined as:
Also, \({q^{b}}\)-definite integral on [a, b] is defined as:
When \(a=0\) in (9), we get Jackson integral defined in (8).
Definition 10
[15] Let \(f:[a,b]\rightarrow \mathbb {R}\) be a continuous function, \(x\in [a,b]\), \(0<q<1\) and \(h\in \mathbb {R}\). Then \( (q_{a}-h) \)-integral on [a, b] is defined as:
whereas, \(({q^{b}-h)}\)-integral on [a, b] is defined as:
When \(h=0\) in Definition (10), we get integrals defined in (9). Let us now recall the results on quantum Hermite–Hadamard integral inequalities:
Theorem 3
[3] Let \(0<q<1\), and \(f:[a,b]\rightarrow \mathbb {R}\) a differentiable convex function on an interval (a, b). Then, the q -Hermite–Hadamard inequality on \(q_{a}\)-integral is given as:
Theorem 4
[3] Let \(0<q<1\), and \(f:[a,b]\rightarrow \mathbb {R}\) a differentiable convex function on an interval (a, b), the q-Hermite–Hadamard inequality on \(q_{a}\)-integral is given by:
Theorem 5
[3] Let \(0<q<1\), and \(f:[a,b]\rightarrow \mathbb {R}\) a differentiable convex function on (a, b), the q-Hermite–Hadamard inequality on \(q_{a}\) -integral is given as:
Furthermore, the q-Hadamrd inequality for convex functions was established for \(q_{b}\)-integral.
Theorem 6
[4] Let \(f:[a,b]\rightarrow \mathbb {R}\) be a differentiable convex function on an interval (a, b). Then, the q-Hadamard inequality on \(q_{b}\) -integral is given as:
The aforementioned quantum integral inequalities have further generalized forms (see [1, 15, 18, 24]).
2 A genaralization of qauntum calculus
In this section, we define \((\phi -h)\)-derivative and \((\phi -h)\)-integral.
Definition 11
Let \(f:{I}\rightarrow \mathbb {R}\) be a continuous function, \(\Omega \) be an interval containing (0, 1) and \(\phi :\Omega \rightarrow \Omega \) be a function such that \(\phi (x)(x+h)\in {I}\) for all \( x\in \Omega \) and \(h\in \mathbb {R}\). Then \((\phi -h)\)-derivative is defined as follows:
where \(\phi (x)\in (0,1)\). If \(x=u\), then we have \(_{h}D_{\phi }(f(u))=\lim \limits _{x\rightarrow u}(_{h}D_{\phi }(f(x)))\).When \(h=0\) in (2.1), then we have the concept of \(\phi \)-derivative given as follows:
When \(\phi (x)=q\) and \(h=0\) in Definition 11, then we have q-derivative given in Definition 6, When \(\phi (x)=q\) in he Definition 11, then we have \((q-h)\)-derivative presented in Definition 7.
Example 1
Let \(f(x)=x^{n}\), where n is positive integer. Take, \(\phi (x)=x^{2}\), \( \forall x\in (0,1)\). Then
By taking \(h=0\), we have
If \(\phi (x)=q\), then we have \(D_{q}(x)^{n}=[n]_{q}x^{n-1}\), \(q\in (0,1)\) where \([n]_{q}=\frac{1-q^{n}}{1-q}.\)
Definition 12
Let \(f:[a,b]\rightarrow \mathbb {R}\) be a continuous function, \( \Omega \) an interval containing (0, 1) and \(\phi :\Omega \rightarrow \Omega \) be a function such that \(\phi (x)(x+h)\in {I}\), \(\forall x\in \Omega \), \( h\in \mathbb {R}\), and \(\phi (x)\in (0,1)\). The left \((\phi -h)\)-integral is defined as:
and the right \((\phi -h)\)-integral is defined as:
When \(\phi (x)=q\) and \(h=0\) in Definition 12, then we have Definition 9, When \(\phi (x)=q\) in Definition 12, then we have Definition 10.
Example 2
If \(0<\phi (x)<1\), and \(f(x)=x\) for all \(x\in [a,b]\), then we have
and
If we take \(f(x)=1\) in Example 2, we get:
Example 3
Let \(k\in \mathbb {N}\), \(\phi (x)=x\) and \(f(x)=x^{k}\), for all \( x\in [a,b]\), then
where
If we take \(\phi (x)=x\), \(x\in (0,1)\) and \(k=3\), then we obtain that
where
Hence, we have
If \(h=0\), then we get
As a special case, if we take \(a=0\) and \(\phi (x)=x=q\), then we have
In the next section, we give new extensions of the q-Hermite–Hadamard inequality. Here, we will employ \((\phi -h)\)-integrals to demonstrate inequalities for convex, m-convex, and \(\hbar \)-convex functions. Additionally, we will discuss theorems that have been proved using q-integrals.
3 Extended forms of the q-Hermite–Hadamard inequalities
Theorem 7
Let \(f:[a,b]\rightarrow \mathbb {R}\) be a convex function, \(\Omega \) an interval containing (0, 1) and \(\phi :\Omega \rightarrow {\Omega }\) a function such that \(\phi (x)(x+h)\in [a, b]\), \(\forall x\in {\Omega }\), \( \text {and}\,h\in \mathbb {R}\). Then the following inequality holds for \((\phi -h)\)-integrals:
Proof
Using the Jensen’s inequality, we have
where \(\sum _{n=0}^{\infty }(1-\phi (x))\phi ^{n}(x)=1\).On simplifying the left hand side and applying the definition of left \((\phi -h)\)-integral on right hand side of the inequality, we obtain that
Again, by using the Jensen’s inequality, we get
On simplification the left side of the above inequality and using the definition of right \((\phi -h)\)-integral on the right side, we get
Note that, the left inequality of (3.1) is obtained by adding (3.3 ) and (3.5). Let Z(x) be the chord joining two points (a, f(a)) and (b, f(b)). As f is a convex function, \(f(x)\le Z(x)\) which can be stated as:
That is,
The left \((\phi -h)\) integration of (3.7) over the interval [a, x) gives
which takes the following form:
Moreover, the right \((\phi -h)\) integration of (3.7) over the interval (x, b] gives
which takes the following form:
By combining (3.9) and (3.11), we obtain the second inequality of ( 3.1). \(\square \)
Corollary 1
Take \(h=0\) in (3.1) and obtain the inequality for \(\phi \)- integrals:
Remark 1
Setting \(x=b\) in (3.3), \(x=a\) in (3.5), and \(x=b\) in ( 3.9), \(x=a\), in (3.11), we get the following inequalities:
Corollary 2
Taking \(h=0\) in the Remark 1, the Hermite–Hadamard inequality for \( \phi \)-integrals is given below:
If we set, \(\phi (x)=q\), we get q-Hermite–Hadamard inequality which reduces classical Hermite–Hadamard inequality on taking \(q=1\).
Using the Hermite–Hadamard inequality for \(\hbar \)-convex functions, we can now demonstrate the following generalization.
Theorem 8
Let \(f:[a,b]\rightarrow \mathbb {R}\) be \(\hbar \)-convex function such that \(\hbar (\frac{1}{2})\ne 0\), \(\Omega \) an interval containing (0, 1) and \(\phi :\Omega \rightarrow {\Omega }\) be a function such that \( \phi (x)(x+h)\in [a, b]\), \(\forall x\in {\Omega }\), \(\text { and}\,\) \(h\in \mathbb {R}\). Then the following inequality holds for \((\phi -h)\)-integral. (i) The following inequality is applicable to the left \((\phi -h)\) -integral if f is symmetric about \(\frac{a+x}{2},\forall x\in (a,b)\).
(ii) The following inequality is applicable to the right \((\phi -h)\) -integral if f is symmetric about \(\frac{x+b}{2},\forall x\in (a,b)\).
Proof
We demonstrate (i) and (ii) in the following way:
(i) Since f is \(\hbar \)-convex function, we get:
Applying the \((\phi -h)\)-integral on both sides over the interval [0, 1]:
Since \(f(a+x-u)=f(u)\), \(\forall u\in (a,x)\), we have \(f(x-\lambda (x-a))=f(a+\lambda (x-a))\). Then, the inequality (3.18) takes the following form:
Applying the left \((\phi -h)\)-integral definition, we get:
Now, using the \(\hbar \)-convexity of f on the right hand side of the equality (3.20), we have
Therefore, (3.20) becomes
Finally, we obtain the inequality (3.15) from (3.19), (3.20) and (3.21):(ii) By using the \(\hbar \)-convexity, we get
Applying the \((\phi -h)\)-integral on the both sides over the interval [0, 1], we have
Since \(f(x+b-u)=f(u)\), \(\forall u\in (x,b),\) we have \(f(b-\lambda (b-x))=f(x+\lambda (b-x))\). The inequality (3.23) becomes:
We now apply the right \((\phi -h)\)-integral to obtain the following
As, f is \(\hbar \)-convex, the right hand side of equality (3.25) becomes
So, (3.25) reduces to
We obtain the inequality (3.16) from (3.24), (3.25) and (3.26). \(\square \)
Corollary 3
Taking \(h=0\) in (3.15) and (3.16), we obtain the inequalities for the left and right \(\phi \)-integrals as follows:
and
Remark 2
By setting \(x=b\) in (3.15) and \(x=a\) in (3.16), we get the following inequality:
where \( h_{3}=(b-a)h. \)
Corollary 4
Using \(h=0\) in the Remark 2, the inequality for \({\phi } \) -integrals is given as:
Theorem 9
Let \(f:[0,y]\rightarrow \mathbb {R}\) be m-convex function, \( \Omega \) an interval containing (0, 1) and \(\phi :\Omega \rightarrow { \Omega }\) be a function such that \(\phi (x)(x+h)\in [0, y]\), \(\forall x\in { \Omega }\), \(h\in \mathbb {R}\) and \(a,b\in [0,y]\) with \(a<b\). Then,
(i) The following inequality holds for the left \((\phi -h)\) -integral provided that \(f(\frac{a+x-u}{m})=f(u),\forall {u}\in (a,x)\)
(ii) The following inequality holds for the right \((\phi -h)\) -integral if \(f(\frac{x+b-u}{m})=f(u),\forall {u}\in (x,b)\)
Proof
We now prove (i) and (ii) as follows:
(i) It follows from the m-convexity of f that
Using the fact: \(f(\frac{x+a-u}{m})=f(u),\forall {u}\in (a,x)\), we have \( f\left( \frac{x-\lambda (x-a)}{m}\right) =f(a+\lambda (x-a))\). The above inequality becomes
Applying the left \((\phi -h)\)-integral on the both sides over the interval [0, 1], we have
By using the \(\hbar \)-convexity of f, the right term of inequality (3.33) reduces to
From (3.33), (3.20) and (3.34), we obtain the following:
On \((\phi -h)\)-integrating (3.35) over the interval [0, 1], we get the inequality (3.29). Following arguments similar to those given above, we obtain (3.30) inequality from (3.33) by using the definition of right \((\phi -h)\) -integral.
\((\phi -h)\)-integrating (3.36) over the interval [0, 1] gives the inequality (3.30). \(\square \)
Corollary 5
Taking \(h=0\) in (3.29) and (3.30), we obtain the inequalities for the left and right \(\phi \)-integrals as follows:
Remark 3
By setting \(x=b\) in (3.29) and \(x=a\) in (3.30), we get the following inequality:
where \( h_{3}=(b-a)h. \)
Corollary 6
If we take \(h=0\) in (3.40), then we have the inequality for \(\phi \) -integral:
where \( h_{3}=(b-a)h. \)
Remark 4
If we take \(\phi (x)=q\); constant function, then the results in [7] are obtained.
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Akbar, S.B., Abbas, M. & Budak, H. Generalization of quantum calculus and corresponding Hermite–Hadamard inequalities. Anal.Math.Phys. 14, 99 (2024). https://doi.org/10.1007/s13324-024-00960-9
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DOI: https://doi.org/10.1007/s13324-024-00960-9
Keywords
- (\({\phi }-h)\)-derivative
- \(({\phi }-h)\)-integral
- Jensen inequality
- \(\hbar \)-convex function
- m-convex function
- Hermite Hadamard Inequality