Abstract
We use a new function class called B-function to establish a novel version of Hermite–Hadamard inequality for weighted ψ-Hilfer operators. Additionally, we prove two new identities involving weighted ψ-Hilfer operators for differentiable functions. Moreover, by employing these equalities and the properties of the B-function, we derive several trapezoid- and midpoint-type inequalities for h-convex functions. Furthermore, the obtained results are reduced to several well-known and some new inequalities by making specific choices of the function h.
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1 Introduction & preliminaries
In recent decades, various publications have focused on generalizing the Hermite–Hadamard inequality and developing trapezoid- and midpoint-type inequalities that provide bounds for the right- and left-hand sides of the aforementioned inequality. The authors [11] demonstrated various similar trapezoid-type inequalities and developed the Hermite–Hadamard inequality for Riemann–Liouville fractional integrals. Kara et al. [8] identified the following Hermite–Hadamard inequalities:
Let \(\psi :[a,b]\rightarrow \mathbb{R}\) be a monotone increasing function such that the derivative \(\psi ^{\prime }>0\) is continuous on \((a,b)\). If g is a convex function on \([a,b]\), then
where the ψ-Hilfer operators are defined as follows:
and
See [3, 7, 9, 12] for further information on comparable results.
In [13], the author introduces a novel class of functions, called h-convex functions.
Definition 1
Let \(h : J \subseteq \mathbb{R} \rightarrow \mathbb{R}\), where \((0,1) \subseteq J \), be a nonnegative function, \(h \neq 0\). We say that \(f : I\subseteq \mathbb{R} \rightarrow \mathbb{R}\) is an h-convex function if f is nonnegative and for all \(x, y \in I\), \(\alpha \in (0, 1)\) we have
If the inequality in (1.2) is reversed, then f is said to be h-concave.
By setting
-
\(h(\lambda )=\lambda \), Definition 1 reduces to that of the classical convex function.
-
\(h(\lambda )=1\), Definition 1 reduces to that of P-functions [4, 10].
-
\(h(\lambda )=\lambda ^{s}\), Definition 1 reduces to that of s-convex functions [2].
-
\(h(\lambda )=\frac{1}{n}\sum_{k=1}^{n}\lambda ^{ \frac{1}{k}}\), Definition 1 reduces to that of polynomial n-fractional convex functions [5].
Recently, the authors of [1] presented a new class of function, called B-function.
Definition 2
Let \(a< b\) and \(g : (a, b)\subset \mathbb{R} \rightarrow \mathbb{R}\) be a nonnegative function. The function g is a B-function, or g belongs to the class \(B(a, b)\), if for all \(x\in (a, b)\), we have
If the inequality (1.3) is reversed, g is called an A-function, or we say that g belongs to the class \(A(a, b)\).
If we have the equality in (1.3), then g is called an AB-function, or we say that g belongs to the class \(AB(a, b)\).
Corollary 1
Let \(h : (0, 1) \rightarrow \mathbb{R}\) be a nonnegative function. The function h is a B-function if and only if for all \(\lambda \in (0, 1)\), we have
-
The functions \(h(\lambda )=\lambda \) and \(h(\lambda )=1\) are AB-functions, B-functions, and A-functions.
-
The function \(h (\lambda )=\lambda ^{s}\), \(s\in (0, 1]\) is a B-function.
-
The function \(h(\lambda ) = \frac{1}{n}\sum _{k=1}^{n}\lambda ^{\frac{1}{k}}, n\), \(k\in \mathbb{N}\) is a B-function.
The weighted fractional integrals are defined as follows:
Definition 3
([6])
Let \([a,b]\subseteq {}[ 0,+\infty )\). Let \(\beta >0\) and ψ be a positive, increasing differentiable function such that \(\psi ^{\prime }(s)\neq 0\) for all \(s\in {}[ a,b]\). The left- and right-sided weighted fractional integrals of a function f with respect to the function ψ on \([a,b]\) are respectively defined as follows:
where w is a weighted function and the gamma function defined by
For these operators, consider the following space:
For special choices of ψ, w, and β, we get already known results.
-
(1)
Taking \(w(t)=1\), the operators reduce to the ψ-Hilfer integral operators of order \(\beta >0\).
-
(2)
For \(\psi (t)=t \), we get the weighted Riemann–Liouville operators.
-
(3)
For \(\psi (t)=t \) and \(w(t)=1\), the operators are simplified to Riemann–Liouville integral operators.
-
(4)
Taking \(\psi (t)=t \), \(w(t)=1\), and \(\beta = 1 \), the operators reduce to classical Riemann integrals.
-
(5)
Setting \(\psi (t)= \ln (t) \) and \(a>1\), we get the weighted Hadamard operators of order \(\beta > 0\).
-
(6)
Setting \(\psi (t)= \ln (t) \), \(w(t)=1\), and \(a>1\), the operators are simplified to Hadamard operators of order \(\beta > 0\).
The purpose of this study is to generalize the Hermite–Hadamard inequality given in [8] for the h-convex function and weighted ψ-Hilfer operator with conditions. For this aim, we assume h is a B-function.
2 Hermite–Hadamard inequality
This section establishes Hermite–Hadamard-type inequalities for h-convex functions using ψ-Hilfer operators. Throughout this paper, we consider that \(0\leq a< b<\infty \), \(\beta >0\), and ψ is a positive differentiable increasing function on \((a,b)\).
Theorem 2.1
Let h be a B-function and w a nondecreasing function. If \(f\in X [a,b]\) is an h-convex function, then the following inequalities hold:
where
and
Proof
Since w is a positive nondecreasing function on \([a,b]\),
-
(1)
for all \(\tau \in [a,\frac{a+b}{2}]\), we have \(0 < w(a)\leq w(\tau )\leq w(\frac{a+b}{2})\leq w(b)\), and then
$$\begin{aligned} \begin{aligned} \frac{w(a)}{\beta} \biggl(\psi \biggl( \frac{a+b}{2} \biggr)-\psi ( a) \biggr) ^{\beta} & \leq \int _{a}^{\frac{a+b}{2}} \biggl(\psi \biggl( \frac{a+b}{2} \biggr)-\psi (\tau ) \biggr)^{\beta -1}w(\tau ) \psi '( \tau )\,d\tau \\ & \leq \frac{w(b)}{\beta} \biggl(\psi \biggl( \frac{a+b}{2} \biggr)-\psi (a) \biggr) ^{\beta};\end{aligned} \end{aligned}$$(2.4) -
(2)
for all \(\tau \in [\frac{a+b}{2},b]\), we have \(0 < w(a)\leq w(\frac{a+b}{2})\leq w(\tau )\leq w(b)\), and then
$$ \begin{aligned} \frac{w(a)}{\beta} \biggl(\psi (b)- \psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta} & \leq \int _{\frac{a+b}{2}}^{b} \biggl(\psi (\tau )-\psi \biggl( \frac{a+b}{2} \biggr) \biggr)^{ \beta -1}w(\tau ) \psi '(\tau ) \,d\tau \\ & \leq \frac{w(b)}{\beta} \biggl(\psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta}.\end{aligned}$$(2.5)
Letting f be an h-convex function, we have for any \(\tau \in {}[ a,b]\),
and then
Multiplying (2.6) by \(( \psi (\frac{a+b}{2})-\psi (\tau ) ) ^{\beta -1}\psi '( \tau )w(\tau )\) and integrating over \(\tau \in {}[ a,\frac{a+b}{2}]\), we obtain
By using the left-hand side of (2.4), we deduce
Now, multiplying (2.6) by \(( \psi (\tau )-\psi ( \frac{a+b}{2} ) ) ^{ \beta -1}\psi '(\tau )w(\tau )\) and integrating over \(\tau \in [ \frac{a+b}{2},b ] \), we get
By using the left-hand side of (2.5), we deduce
Adding the inequalities (2.7) and (2.8), we obtain
Let us prove the second inequality in (2.1). Since any \(\tau \in [ a,b ] \) can be written as \(\tau =(1-t)a+tb \) for \(t\in [ 0,1 ] \), we have
Applying the h-convexity of the function f, we get
Applying (1.4), we deduce
Multiplying (2.10) by \(( \psi (\frac{a+b}{2})-\psi (\tau ) ) ^{\beta -1}\psi '( \tau )w(\tau )\) and integrating over \(\tau \in {}[ a,\frac{a+b}{2}]\), we obtain
By using the right-hand side of (2.4), we deduce
Now, multiplying (2.10) by \(( \psi (\tau )-\psi ( \frac{a+b}{2} ) ) ^{ \beta -1}\psi '(\tau )w(\tau )\) and integrating over \(\tau \in [ \frac{a+b}{2},b ] \), we get
By using the right-hand side of (2.5), we deduce
Adding inequalities (2.11) and (2.12), we obtain
This finishes the proof. □
The following results are dependent on the function h presented in Theorem 2.1. First, assuming \(h(\alpha )=\alpha \), we get the following result using the weighted ψ-Hilfer operators for convex functions.
Corollary 2
Let \(f\in X [a,b]\) be a convex function. Then the following inequalities hold:
where \(F( t)\) and \(\Omega (\psi ,\beta ) \) are defined by (2.2) and (2.3), respectively.
By setting \(h(\alpha )=1\), we get the following result using the weighted ψ-Hilfer operators with an f being a P-function.
Corollary 3
Let \(\beta >0\) and \(f\in X [a,b]\) be a P-function. Then the following inequalities hold:
where \(F( t)\) and \(\Omega (\psi ,\beta ) \) are defined by (2.2) and (2.3), respectively.
Using \(h(\alpha )=\alpha ^{s}\), we obtain the following result through the weighted ψ-Hilfer operators and s-convex functions.
Corollary 4
Let \(\beta >0\), \(s\in (0, 1]\), and \(f\in X [a,b]\) be an s-convex function. Then the following inequalities hold:
where \(F( t)\) and \(\Omega (\psi ,\beta ) \) are defined by (2.2) and (2.3), respectively.
Taking \(h(\alpha )=\frac{1}{n}\sum_{k=1}^{n}\alpha ^{\frac{1}{k}}\), we deduce the following result through the weighted ψ-Hilfer operators and n-fractional polynomial convex functions.
Corollary 5
Let \(\beta >0\) and \(f\in X [a,b]\) be an n-fractional polynomial convex function. Then the following inequalities hold:
where \(F( t)\), \(\Omega (\psi ,\beta ) \) are defined by (2.2), (2.3), respectively, and \(C_{n}=\frac{2}{n}\sum_{k=1}^{n} (\frac{1}{2} )^{\frac{1}{k}}\).
Remark 1
If we choose \(\psi (\tau )=\tau \) and \(\psi (\tau )=\ln \tau \) in Corollaries 3, 4, and 5, we obtain Hermite–Hadamard inequality for P-functions, s-convex functions, and n-fractional polynomial convex functions involving the weighted Riemann–Liouville fractional operator and the weighted Hadamard fractional operator, respectively.
3 Weighted trapezoid-type inequalities
This section presents weighted trapezoid inequalities and their particular results utilizing weighted ψ-Hilfer operators with w being symmetric with respect to \(\frac{a+b}{2}\) (i.e., \(w(t)=w(b+a-t)\)). To accomplish this, we must first establish an equality in the following lemma.
Lemma 3.1
Assume w is a differentiable and symmetric with respect to \(\frac{a+b}{2}\) function, and suppose h is a B-function. Let \(f:[a,b]\rightarrow \mathbb{R}\) be a function where \((wf)\) is a differentiable mapping on \((a,b)\). Then the following identity holds:
where
Proof
Let
Integrating by parts (3.4) and using (2.2), we get
Therefore
Similarly, let
Integrating by parts (3.6), we obtain
Since \(F(a)=F(b)=f(a)+f(b)\), we conclude from (3.5) and (3.7) that
thus
On the other hand, since \(F^{\prime }(\tau )=f^{\prime }(\tau )-f^{\prime }(a+b-\tau )\) and \(w(\tau )=w(a+b-\tau )\), we get
From (3.4), we get
By changing the variable \(\tau =\frac{1+s}{2}a+\frac{1-s}{2}b\), we obtain
Similarly, from (3.6) we deduce
Consequently,
Finally, we acquire the needed equality (3.1) by substituting (3.9) into (3.8). □
Remark 2
Putting \(w=1\) in Lemma 3.1, we get [8, Lemma 3.1].
Theorem 3.1
Under the hypotheses of Lemma 3.1, if \(|(wf)^{\prime }|\) is an h-convex mapping on \([a, b]\) and h is a B-function, then the trapezoid-type inequality holds, namely
Proof
Taking the absolute value of the identity (3.1) and using the h-convexity of the function \(|(wf)^{\prime }|\), we get
Given that h is a B-function, setting \(\alpha =\frac{1-s}{2}\) and \(1-\alpha =\frac{1+s}{2}\) yields
□
The following results are obtained via the weighted ψ-Hilfer operators and depend on the function h given in Theorem 3.1.
Corollary 6
-
(1)
If \(|(wf)^{\prime }|\) is a convex mapping on \([a,b]\), then
$$ \begin{aligned} &\biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w ( \frac{a+b}{2} ) }{2 \Phi (\psi ,\beta ,w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{b-a}{4 \Phi (\psi ,\beta ,w)} \bigl[ \bigl\vert (wf)^{ \prime }(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert A_{\psi , \beta }(s) \bigr\vert \,ds.\end{aligned}$$Particularly, putting \(w=1\), we get [8, Corollary 3.4].
-
(2)
If \(|(wf)^{\prime }|\) is a P-function on \([a, b]\), then
$$ \begin{aligned} &\biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w (\frac{a+b}{2} )}{2 \Phi (\psi ,\beta , w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{b-a}{2 \Phi (\psi ,\beta , w)} \bigl[ \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert \,ds.\end{aligned}$$ -
(3)
If \(|(wf)^{\prime }|\) is an s-convex mapping on \([a, b]\), then
$$ \begin{aligned} &\biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w (\frac{a+b}{2} )}{2 \Phi (\psi ,\beta , w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{b-a}{2^{s+1} \Phi (\psi ,\beta , w)} \bigl[ \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert \,ds.\end{aligned}$$ -
(4)
If \(|(wf)^{\prime }|\) is an n-fractional polynomial convex mapping on \([a, b]\), then
$$ \begin{aligned}& \biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w (\frac{a+b}{2} )}{2 \Phi (\psi ,\beta , w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a) C_{n}}{4 \Phi (\psi ,\beta , w)} \bigl[ \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert \,ds,\end{aligned}$$where \(\Phi (\psi ,\beta , w)\), \(A_{\psi ,\beta }(s)\) are defined by (3.2), (3.3), respectively, and \(C_{n}=\frac{2}{n}\sum_{k=1}^{n} (\frac{1}{2} )^{\frac{1}{k}}\).
Theorem 3.2
Let \(p>1\) and \(\frac{1}{p'}+\frac{1}{p}=1\). If \(|(wf)^{\prime} |^{p}\) is an h-convex mapping on \([a,b]\), then
Proof
Taking absolute value of (3.1) and using the well-known Hölder’s inequality, we obtain
Notice that for \(p>1\), \(A,B\geq 0\), \(A^{\frac{1}{p}}+B^{\frac{1}{p}}\leq 2^{1-\frac{1}{p}}(A+B)^{\frac{1}{p}}\), and \(|(wf)^{\prime } |^{p}\) an h-convex function, we get
Since h is a B-function, we get
This proves the first inequality in (3.11).
Notice that the inequality \(A^{p}+B^{p}\leq (A+B)^{p}\) yields the second inequality in (3.11). □
Setting \(w=1\) and \(h(s)=s\) in Theorem 3.2, we get the following corollary.
Corollary 7
Let \(p>1\) and \(\frac{1}{p^{\prime }}+\frac{1}{p}=1\). If \(|f^{\prime} |^{p}\) is a convex mapping on \([a,b]\), then
which is a better estimate compared with [8, Theorem 3.5].
4 Weighted midpoint-type inequalities
This section establishes some weighted midpoint inequalities for weighted ψ-Hilfer operators using the identity in the following lemma.
Lemma 4.1
Under the hypothesis of Lemma 3.1, the following identity holds:
where \(\Omega (\psi ,\beta )\) and \(A_{\psi , \beta}(\tau )\) are defined in (2.3) and (3.3), respectively.
Proof
Let
By using (3.4), we get
Applying (3.5), we obtain
Similarly, let
Using (3.6), then we have
and applying (3.7), we get
In addition, according to (4.2),
Similarly, from (4.4) we get
As a result,
To obtain the desired equality (4.1), substitute (4.7) into (4.6). □
Remark 3
Put \(w=1\) in Lemma 4.1, we get [8, Lemma 4.1].
Theorem 4.1
If \(|(wf)^{\prime }|\) is an h-convex mapping on \([a, b]\) and h is a B-function, then
Proof
Taking the absolute value of the identity (4.1) and using the h-convexity of \(|(wf)^{\prime }|\) and inequality (1.4), we deduce
This ends the proof. □
The following results are obtained using the weighted ψ-Hilfer operators and depend on the function h given in Theorem 4.1.
Corollary 8
-
(1)
If \(|(wf)^{\prime }|\) is a convex mapping on \([a,b]\), then
$$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{b-a}{4 \Omega (\psi ,\beta ) w ( \frac{a+b}{2} ) } \bigl[ \bigl\vert (wf)^{\prime }(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert \Omega ( \psi ,\beta )-A_{\psi ,\beta }(s) \bigr\vert \,ds.\end{aligned}$$Particularly, putting \(w=1\), we get [8, Theorem 4.2].
-
(2)
If \(|(wf)^{\prime }|\) is a P-function on \([a, b]\), then
$$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ { \mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{b-a}{2 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \bigl[ \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert \Omega ( \psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert \,ds.\end{aligned}$$ -
(3)
If \(|(wf)^{\prime }|\) is an s-convex mapping on \([a, b]\), then
$$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ { \mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{b-a}{2^{s+1} \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \bigl[ \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert \Omega ( \psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert \,ds.\end{aligned}$$ -
(4)
If \(|(wf)^{\prime }|\) is an n-fractional polynomial convex mapping on \([a,b]\), then
$$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{(b-a) C_{n}}{4 \Omega (\psi ,\beta ) w ( \frac{a+b}{2} ) } \bigl[ \bigl\vert (wf)^{\prime }(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr]\int _{0}^{1} \bigl\vert \Omega (\psi ,\beta )-A_{\psi ,\beta }(s) \bigr\vert \,ds,\end{aligned}$$where \(\Omega (\psi ,\beta )\), \(A_{\psi ,\beta }(s)\) are defined by (2.3), (3.3), respectively, and \(C_{n}=\frac{2}{n}\sum_{k=1}^{n} ( \frac{1}{2} ) ^{\frac{1}{k}}\).
Theorem 4.2
Let \(p>1\) and \(\frac{1}{p^{\prime }}+\frac{1}{p}=1\). If \(|(wf)^{\prime} |^{p}\) is an h-convex mapping on \([a,b]\), then
Proof
Taking the absolute value of (4.1) and using the well-known Hölder’s inequality, we obtain
Noticing that \(A^{\frac{1}{p}}+B^{\frac{1}{p}}\leq 2^{1-\frac{1}{p}}(A+B)^{\frac{1}{p}}\) and \(|(wf)^{\prime } |^{p}\) is an h-convex function, we conclude
Putting \(\alpha =\frac{1-s}{2}\) and \(1-\alpha =\frac{1+s}{2}\) yields
This proves the first inequality in (4.9).
The second inequality in (4.9) is clear from the inequality \(A^{p}+B^{p}\leq (A+B)^{p}\). □
Setting \(w=1\) and \(h(s)=s\) in Theorem 4.2, we get the following corollary.
Corollary 9
Let \(p>1\) and \(\frac{1}{p^{\prime }}+\frac{1}{p}=1\). If \(|f^{\prime} |^{p}\) is a convex mapping on \([a,b]\), then
which is a better estimate compared with [8, Theorem 4.5].
5 Conclusions
In this study, we recalled a new function class, namely that of B-functions, and utilized it to derive a novel version of the Hermite–Hadamard inequality for weighted ψ-Hilfer operators. We also established two new identities involving weighted ψ-Hilfer operators for differentiable functions. By combining these identities and the properties of the B-function, we obtained several trapezoid- and midpoint-type inequalities for h-convex functions. Our results not only extend the existing literature on inequalities involving fractional operators but also provide new insights into the behavior of h-convex functions under these operators. Additionally, our methods can be applied to other fractional integral operators by using B-functions.
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The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.
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B. B. and N. A. wrote the main results. H. B. revised the paper.
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Benaissa, B., Azzouz, N. & Budak, H. Weighted fractional inequalities for new conditions on h-convex functions. Bound Value Probl 2024, 76 (2024). https://doi.org/10.1186/s13661-024-01889-5
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DOI: https://doi.org/10.1186/s13661-024-01889-5