Abstract
We employ a new function class called B-function to create a new version of fractional Hermite–Hadamard and trapezoid type inequalities on the right-hand side that involves h-convex and \(\psi \)-Hilfer operators. We also provide new midpoint-type inequalities using h-convex functions.
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1 Introduction
Convexity theory provides robust approaches and concepts for solving an extensive variety of problems in pure and practical mathematics. Because of their tenacity, convex functions have been applied to a wide range of mathematical disciplines, resulting in the discovery of several inequalities in the literature. One famous inequality is the Hermite–Hadamard inequality, which has found applications in convexity theory. In [16], the author introduces a novel class of functions called as h-convex functions
Definition 1
Let \(h:J\rightarrow \mathbb {R}\) be a non-negative function, \( h\ne 0\). We say that \(f:I\rightarrow \mathbb {R}\) is an h-convex function, if f is non-negative and for all \(x,y\in I\), \(\alpha \in (0,1)\) we have
If the inequality (1.1) is reversed, then f is said to be h-concave.
By putting \(h(\alpha )=\alpha \), \(h(\alpha )=1\), \(h(\alpha ) =\alpha ^{s}\), \(h(\alpha )=\displaystyle \frac{1}{n} \sum \nolimits _{k=1}^{n}\alpha ^{\frac{1}{k}}\) and \(h(\alpha ) =\frac{1}{\alpha }\) in (1.1), Definition 1 reduces to some wel known function classes, P-functions [5, 11], s-convex functions [3], n-fractional polynomial convex functions [7] and Godunova–Levin functions [6], respectively.
2 \(\psi \)-Hilfer operators
Definition 2
Let \([a,b]\subseteq [ 0,+\infty ).\) Let \(\beta >0\) and \(\psi \) be a positive, strictly increasing differentiable function such that \(\psi \,^{\prime }(\tau )\ne 0\) for all \(\tau \in [ a,b]\). The left and right sided \(\psi \)-Hilfer fractional integral of a function f with respect to the function \(\psi \) on [a, b] are defined respectively as follows.
where the gamma function verified
For these operators, consider the following space
One essential property of \(\psi \)-Hilfer operators is that they are dependent on the function \(\psi \) and produce a particular type of fractional integrals.
-
(1)
Taking \(\psi (\tau )=\tau \), we get Riemann-Liouville fractional operator of order \(\beta >0\)
$$\begin{aligned}{} & {} \mathcal{R}\mathcal{L}_{a^{+}}^{\beta }f(x)=\displaystyle \frac{1}{\Gamma (\beta )} \int _{a}^{x}(x-s)^{\beta -1}f(s)ds,\quad x>a, \\{} & {} \mathcal{R}\mathcal{L}_{b^{-}}^{\beta }f(x)=\displaystyle \frac{1}{\Gamma (\beta )} \int _{x}^{b}(s-x)^{\beta -1}f(s)ds,,\quad x<b. \end{aligned}$$ -
(2)
Using \(\psi (\tau )=\ln \tau \), we deduce Hadamard fractional operator of order \(\beta >0\)
$$\begin{aligned}{} & {} \mathcal {H}_{a^{+}}^{\beta }f(x)=\displaystyle \frac{1}{\Gamma (\beta )} \int _{a}^{x}\left( \ln \frac{x}{s}\right) ^{\beta -1}f(s)\frac{ds}{s}, \quad x>a>1, \\{} & {} \mathcal {H}_{a^{+}}^{\beta }f(x)=\displaystyle \frac{1}{\Gamma (\beta )} \int _{x}^{b}\left( \ln \frac{s}{x}\right) ^{\beta -1} f(s)\frac{ds}{s},\quad 1<x<b. \end{aligned}$$ -
(3)
Putting \(\psi (\tau )=\frac{\tau ^{\rho }}{\rho }\) where \(\rho >0\), we obtain Katugompola fractional operators of order \(\beta >0\).
$$\begin{aligned}{} & {} \mathcal {K}_{a^{+}}^{\beta }f(x)=\displaystyle \frac{(\rho )^{1-\beta }}{\Gamma (\beta )}\int _{a}^{x}\left( x^{\rho }-s^{\rho }\right) ^{\beta -1} f(s)s^{\rho -1}ds,\ x>a, \\{} & {} \mathcal {K}_{b^{-}}^{\beta }f(x)=\displaystyle \frac{(\rho )^{1-\beta }}{ \Gamma (\beta )}\int _{x}^{b}\left( s^{\rho }-x^{\rho }\right) ^{\beta -1}f(s)s^{\rho -1}ds,\ x<b. \end{aligned}$$
In the literature, papers devoted to fractional integral inequalities. For some of them, please refer to [1, 2, 4, 9, 12,13,14].
3 B-function
We now give a definition of the B-function.
Definition 3
Let \(a<b\) and \(h:(a,b)\subset \mathbb {R}\rightarrow \mathbb {R}\) be a non-negative function. The function h is a B-function, or that h belongs to the class B(a, b), if for all \(x\in (a,b)\), we have
If the inequality (3.1) is reversed, h is called A-function, or that h belongs to the class A(a, b).
If we have the equality in (3.1), h is called AB-function, or that h belongs to the class AB(a, b).
Corollary 1
Let \(h: (0, 1) \rightarrow \mathbb {R}\) be a non-negative function.
-
(1)
The function \(h(\alpha )=1\) is AB-function, B-function and A-function. And the function \(h(\alpha )=\alpha \) is AB-function, B-function and A-function under the conditions \(a=0\), \(a<0\) and \(a>0\), respectively.
-
(2)
The function \(h(\alpha )=\alpha ^{s},\ s\in (0,1]\) is B-function.
-
(3)
The function \(h(\alpha ) =\displaystyle \frac{1}{n}\sum \nolimits _{k=1}^{n} \alpha ^{\frac{1}{k}},\ n,k\in \mathbb {N}\) is B-function.
-
(4)
The function \(h(\alpha )=\frac{1}{\alpha }\) is A-function.
Proof
-
(1)
The first case is obvious.
The following result (3.2) is required to prove the next case.
Let \(g(x)=\ln (x+1)\), \(g(0)=0\), \(g(1)=\ln 2\) and \(M(1,\ln 2)\). On the interval [0, 1], the graph of the function g appears over the line (OM). This gives us, for all \(x\in (0,1]\)
$$\begin{aligned} \ln (x+1)\ge x\ln 2\Leftrightarrow (1+x)^{\frac{1}{x}}\ge 2, \end{aligned}$$taking \(x=\frac{1}{p}\), we have
$$\begin{aligned} \hbox {for}\ \hbox {all}\ p\ge 1, \quad \left( 1+\frac{1}{p}\right) ^{p}\ge 2\,. \end{aligned}$$(3.2) -
(2)
Let \(h(\alpha )=\alpha ^{s},\ s\in (0,1]\) and taking \(s=\frac{1}{p}\). The function h is a B-function, this means
$$\begin{aligned} \alpha ^{\frac{1}{p}}+(1-\alpha )^{\frac{1}{p}} \le \left( \frac{1}{2} \right) ^{\frac{1}{p}-1}. \end{aligned}$$(3.3)We use absurdity to demonstrate inequality (3.3), suppose that exist \(p\ge 1\) verified
$$\begin{aligned} \alpha ^{\frac{1}{p}}+(1-\alpha )^{\frac{1}{p}} >\left( \frac{1}{2}\right) ^{\frac{1}{p}-1}, \end{aligned}$$we get
$$\begin{aligned} \displaystyle \int _{0}^{1}\left[ \alpha ^{\frac{1}{p}} +(1-\alpha )^{\frac{1}{p}}\right] d\alpha >\left( \frac{1}{2}\right) ^{\frac{1}{p}-1}, \end{aligned}$$thus
$$\begin{aligned} 2\displaystyle \left( \frac{p}{p+1}\right) >\left( \frac{1}{2}\right) ^{\frac{1}{p}-1}. \end{aligned}$$This gives
$$\begin{aligned} \displaystyle \left( \frac{p}{p+1}\right) ^{p}>\frac{1}{2}, \end{aligned}$$hence
$$\begin{aligned} \left( 1+\frac{1}{p}\right) ^{p}<2\ \hbox {which}\ \hbox {is}\ \hbox {absurd}. \end{aligned}$$ -
(3)
Let \(h(\alpha )=\displaystyle \frac{1}{n}\sum \nolimits _{k=1}^{n} \alpha ^{\frac{1}{k}},\ n,k\in \mathbb {N}\), we have
$$\begin{aligned} \alpha ^{\frac{1}{k}}+(1-\alpha )^{\frac{1}{k}}\le \left( \frac{1}{2}\right) ^{\frac{1}{k}-1}, \end{aligned}$$then
$$\begin{aligned} \displaystyle \frac{1}{n}\sum \limits _{k=1}^{n}\alpha ^{\frac{1}{k}} +\displaystyle \frac{1}{n}\sum \limits _{k=1}^{n}(1-\alpha )^{\frac{1}{k}} \le \displaystyle \frac{2}{n}\sum \limits _{k=1}^{n}\left( \frac{1}{2} \right) ^{\frac{1}{k}}. \end{aligned}$$This yields
$$\begin{aligned} h(\alpha )+h(1-\alpha )\le 2\,h\left( \frac{1}{2}\right) . \end{aligned}$$ -
(4)
Let \(h(\alpha )=\frac{1}{\alpha }\) with \(\alpha \in (0, 1)\), we have
$$\begin{aligned} \frac{(2\alpha -1)^{2}}{\alpha (1-\alpha )}\ge 0 \Leftrightarrow \frac{1}{\alpha } +\frac{1}{1-\alpha }\ge 4, \end{aligned}$$therefore
$$\begin{aligned} h(\alpha ) +h(1-\alpha )\ge 2 \,h\left( \frac{1}{2}\right) . \end{aligned}$$
\(\square \)
Motivated by previous literature, we use \(\psi \)-Hilfer operators to establish a new version of Hermite–Hadamard and trapezoid inequalities based on h-convex functions where h belongs to the class B(0, 1). We also provide new midpoint inequalities using h-convex functions.
4 Hermite–Hadamard inequalities
We now present the first result of the Hermite–Hadamard inequality for h-convexity of the function f involving \(\psi \)-Hilfer operators.
Theorem 4.1
Let \(\beta >0\), h be a B-function. Let \(f\in X[a,b]\) be a h-convex function and \(\psi \) be a positive, strictly increasing differentiable function such that \(\psi \,^{\prime }(\tau )\ne 0\) for all \( \tau \in [ a,b]\). Then the following inequalities hold
where
Proof
For any \(t\in [ a,b]\), we have
then
Multiplying (4.3) by \(\beta \left( \psi (b)-\psi (t)\right) ^{\beta -1}\psi ^{^{\prime }}(t)\) and integrating over \(t\in [ a,b]\), we result
Multiplying (4.3) by \(\beta \left( \psi (t)-\psi (a)\right) ^{\beta -1}\psi ^{^{\prime }}(t)\) and integrating over \(t\in [ a,b]\), we get
By adding the inequalities (4.4) and (4.5), we deduce
Put \(t=(1-s)\,a+s\,b\) in (4.2) for \(s\in (0,1)\) and using the h-convexity of f, we get
given that h is a B-function, we conclude
Using the same method as previously on (4.7), we obtain
Combining the inequality (4.6) with the inequality (4.8) yields the desired outcome. \(\square \)
Taking \(\psi (x)=x\) and \(\beta =1\), we conclude the following new version of the Hermite–Hadamard inequality, see (Theorem.1 [15]).
Corollary 2
Let h be a B-function. Let \(f\in L[a,b]\) be a h-convex function. Then the following inequalities hold
The following results are dependent on the function h presented in Theorem 4.1. First, assuming \(h(\alpha )=\alpha \), we may obtain the result using the convex functions described in [8].
Corollary 3
Let \(f\in X[a,b]\) be a convex function and \(\psi \) be a positive, strictly increasing differentiable function such that \(\psi \,^{\prime }(\tau )\ne 0\) for all \(\tau \in [ a,b]\), \(\beta >0\). Then the following inequalities hold
where \(F(t)=f(t)+f(a+b-t)\).
By setting \(h(\alpha )=1\), we get the following result using \(\psi \)-Hilfer operators with function f belongs to class P-function.
Corollary 4
Let \(\beta >0\), \(f\in X[a,b]\) be a P-function and \( \psi \) be a positive, strictly increasing differentiable function such that \(\psi \,^{\prime }(\tau )\ne 0\) for all \(\tau \in [ a,b]\). Then the following inequalities hold
where \(F(t)=f(t)+f(a+b-t)\).
Using \(h(\alpha )=\alpha ^{s}\), we obtain the following result through \(\psi \)-Hilfer operators and s-convex functions.
Corollary 5
Let \(\beta >0\), \(s\in (0,1]\) and \(f\in X[a,b]\) be a s-convex function and let \(\psi \) be a positive, strictly increasing differentiable function such that \(\psi \,^{\prime }(\tau )\ne 0\) for all \( \tau \in [ a,b]\). Then the following inequalities hold
where \(F(t)=f(t)+f(a+b-t)\).
Taking \(h(\alpha )=\frac{1}{n}\sum \nolimits _{k=1}^{n}\alpha ^{\frac{1}{k}}\), we deduce the following result through \(\psi \)-Hilfer operators and n-fractional polynomial convex functions.
Corollary 6
Let \(\beta >0\) and \(f\in X [a,b]\) be a n-fractional polynomial convex function and \(\psi \) be a positive, strictly increasing differentiable function such that \(\psi \,^{\prime }(\tau )\ne 0\) for all \( \tau \in [a, b]\). Then the following inequalities hold
where \(F( t) = f(t) + f( a + b - t) \) and \(C_{n}=\frac{2}{n} \sum \nolimits _{k=1}^{n}\left( \frac{1}{2}\right) ^{\frac{1}{k}}\).
Remark 1
If we choose \(\psi (\tau )=\tau \), \(\psi (\tau )=\ln \tau \) and \(\psi (\tau )=\frac{ \tau ^{\rho }}{\rho }\), respectively, in Corollaries 4, 5 and 6, we obtain Hermite–Hadamard inequality for P-function, s-convex functions and n-fractional polynomial convex functions, respectively, involving Riemann-Liouville fractional operator, Hadamard fractional operator, Katugompola fractional operators, respectively.
5 Trapezoid type inequalities
In this section, we will explore certain trapezoid-like inequalities for h-convexity functions utilizing \(\psi \)-Hilfer operators, as well as their particular results, where h belongs to the class B(0, 1). To accomplish this, we must first establish equality in the lemma that follows.
Lemma 5.1
If \(\beta ,\psi \) are defined as in Theorem 4.1 and \( f:[a,b]\rightarrow \mathbb {R}\) is a differentiable mapping to (a, b), then the following identity holds.
where
Proof
Let
Integrating by parts (5.3) and using (4.2), we get
therefore
Similarly, let
Integrating by parts (5.5), we deduce
Since \(F(a)=F(b)=f(a)+f(b)\), from (5.4) and (5.6), we obtain
thus
On the other hand, given that \(F^{\prime }(t)=f^{\prime }(t)-f^{\prime }(a+b-t)\), then from (5.3), we get
Changing the variable \(t=sa+(1-s)b\) for \(s\in [ 0,1]\) produces
Similarly, from (5.5), we get
As a result,
To get the desired equality (5.1), simply replace (5.8) with (5.7). \(\square \)
Theorem 5.1
Assume h is a B-function and \(\beta , \psi \) are defined according to Lemma 5.1. If \(|f^{\prime }|\) is a h-convex mapping on [a, b], then the trapezoid type inequality is obtained as
where \(\Omega (\psi ,\beta )= \left( \psi (b)-\psi (a)\right) ^{\beta }\).
Proof
Using the absolute value of identity (5.1), the h-convexity of the function \(|f^{\prime }|\) and h is a B-function, we get
This completes the proof. \(\square \)
Using \(\psi (x)=x\) and \(\beta =1\), we derive the following new version of the trapezoidal inequality, see ([10, Corollary.1]).
Corollary 7
Let h be a B-function. If \(|f^{\prime }|\) is a h-convex mapping on [a, b], then the following inequalities hold
The following Corollaries depend on the function h given in Theorem 5.1. First, Let’s put \(h(\alpha )=\alpha \) to get a new version using the convex functions proven in [8].
Corollary 8
Assume that \(\beta ,\psi \) are defined as in Lemma 5.1. If \(|f^{\prime }|\) is a convex mapping on [a, b], then
where \(\Omega (\psi ,\beta )=\left( \psi (b)-\psi (a)\right) ^{\beta }\).
Put \(h(\alpha )=1\), we get the following result via \(\psi \)-Hilfer operators with function f belongs to the class P-function.
Corollary 9
Assume that \(\beta , \psi \) are defined as in Lemma 5.1. If \(|f^{\prime }|\) is a P-function on [a, b], thus
where \(\Omega (\psi ,\beta )= \left( \psi (b)-\psi (a)\right) ^{\beta }\).
Put \(h(\alpha )=\alpha ^{s}\), we get the following result via \(\psi \)-Hilfer operators with s-convex functions.
Corollary 10
Assume that \(\beta , \psi \) are defined as in Lemma 5.1. If \(|f^{\prime }|\) is a s-convex mapping on [a, b], then the trapezoid type inequality is obtained as
where \(\Omega (\psi ,\beta )= \left( \psi (b)-\psi (a)\right) ^{\beta }\).
Put \(h(\alpha )=\frac{1}{n}\sum \nolimits _{k=1}^{n} \alpha ^{\frac{1}{k}}\), we get the following result via \(\psi \)-Hilfer operators with n-fractional polynomial convex functions.
Corollary 11
Assume that \(\beta , \psi \) are defined as in Lemma 5.1. If \(|f^{\prime }|\) is a n-fractional polynomial convex mapping on [a, b], yields
where \(\Omega (\psi ,\beta )= \left( \psi (b)-\psi (a)\right) ^{\beta }\) and \(C_{n}=\frac{2}{n}\sum \nolimits _{k=1}^{n} \left( \frac{1}{2}\right) ^{\frac{1}{k}}\).
Remark 2
If we choose \(\psi (\tau )=\tau \), \(\psi (\tau )=\ln \tau \) and \(\psi (\tau )=\frac{ \tau ^{\rho }}{\rho }\), respectively, in the corollaries 8, 9, 10 and 11, we establish trapezoid-type inequalities for the convex function, the function P, s-convex functions, and polynomial n-fractional convex functions, respectively, involving the Riemann-Liouville fractional operator, the Hadamard fractional operator, and the Katugompola fractional operators, respectively.
Theorem 5.2
Let h be a B-function, \(p>1\) and \(\frac{1}{p^{\prime }}+ \frac{1}{p}=1\). Assume that \(\beta ,\psi \) are defined as in Lemma 5.1. If \(\left| f^{\prime }\right| ^{p}\) is a h-convex mapping on [a, b], then the following trapezoid type inequality is obtained
where \(\Omega (\psi ,\beta )=\left( \psi (b)-\psi (a)\right) ^{\beta }\).
Proof
By using the absolute value on equality in Lemma 5.1, we get
Utilizing Hölder’s inequality, we get
Notice that for \(p>1\) and \(A,B\ge 0\): \(\displaystyle A^{\frac{1}{p}}+B^{ \frac{1}{p}}\le 2^{1-\frac{1}{p}}(A+B)^{\frac{1}{p}}\), then
Given that \(\left| f^{\prime }\right| ^{p}\) is h-convex function and h is B-function, we get
therefore
This accomplishes the first inequality in (6.14). Notice that \( A^{p}+B^{p}\le (A+B)^{p}\ \hbox {for}\ p>1\), it results the second inequality in (6.14). \(\square \)
6 Midpoint type inequalities
This section presents new results on midpoint inequality involving h-convex functions via \(\psi \)-Hilfer operators, based on the identity presented in the following lemma.
Lemma 6.1
Let \(\beta >0\), \(f\in X [a,b]\) be a h-convex function and \( \psi \) be a positive, strictly increasing differentiable function such that \( \psi \,^{\prime }(\tau )\ne 0\) for all \(\tau \in [a, b]\). The following identity holds
Proof
Let
Integrating by parts (6.2), we get
This gives
Consider the following integral
Integrating by parts yields
which gives
By adding (6.3) and (6.5), then using (4.2) we obtain
Furthermore, by changing the variable \(t=sa+(1-s)b\) in (6.2) and using (4.2), we get
Changing the variable \(t=(1-s)a+sb\) in (6.4) and using (4.2), we obtain
therefore
Replacing (6.7) in (6.6), we get the desired equality (6.1). \(\square \)
Theorem 6.1
Assume h is a B-function and \(\beta , \psi \) are defined according to Lemma 5.1. If \(|f^{\prime }|\) is a h-convex mapping on [a, b], then the midpoint type inequality is achieve.
Proof
Using the absolute value of the identity (6.1) and the h-convexity of the function \(|f^{\prime }|\), we get
Given that h is a B-function, we get
This end the proof. \(\square \)
Using \(\psi (x)=x\) and \(\beta =1\), we get the following midpoint inequality.
Corollary 12
Assume h is a B-function. If \(|f^{\prime }|\) is a h-convex mapping on [a, b], then the midpoint type inequality holds.
The following Corollaries depend on the function h given in Theorem 6.1. First, Let’s put \(h(\alpha )=\alpha \) to get midpoint inequality with convex functions.
Corollary 13
Assume that \(\beta ,\psi \) are defined as in Lemma 5.1. If \(|f^{\prime }|\) is a convex mapping on [a, b], then
where \(\Omega (\psi ,\beta )=\left( \psi (b)-\psi (a)\right) ^{\beta }\).
Put \(h(\alpha )=1\), we get midpoint type inequality via \(\psi \)-Hilfer operators with function f belongs to the class P-function.
Corollary 14
Assume that \(\beta ,\psi \) are defined as in Lemma 5.1. If \(|f^{\prime }|\) is a P-function on [a, b], thus
where \(\Omega (\psi ,\beta )=\left( \psi (b)-\psi (a)\right) ^{\beta }\).
Put \(h(\alpha )=\alpha ^{s}\), we get the following midpoint type inequality involving \(\psi \)-Hilfer operators with s-convex functions.
Corollary 15
Assume that \(\beta ,\psi \) are defined as in Lemma 5.1. If \(|f^{\prime }|\) is a s-convex mapping on [a, b], then the trapezoid type inequality is obtained as
where \(\Omega (\psi ,\beta )=\left( \psi (b)-\psi (a)\right) ^{\beta }\).
Put \(h(\alpha )=\frac{1}{n}\sum \nolimits _{k=1}^{n}\alpha ^{\frac{1}{k}}\), we get midpoint type inequality via \(\psi \)-Hilfer operators with n-fractional polynomial convex functions.
Corollary 16
Assume that \(\beta ,\psi \) are defined as in Lemma 5.1. If \(|f^{\prime }|\) is a h-fractional polynomial convex mapping on [a, b], yields
where \(\Omega (\psi ,\beta )=\left( \psi (b)-\psi (a)\right) ^{\beta }\) and \( C_{n}=\frac{2}{n}\sum \nolimits _{k=1}^{n}\left( \frac{1}{2}\right) ^{\frac{1}{k} }\).
Remark 3
If we choose \(\psi (\tau )=\tau \), \(\psi (\tau )=\ln \tau \) and \(\psi (\tau )=\frac{ \tau ^{\rho }}{\rho }\), respectively, in the corollaries 13, 14, 15 and 16, we establish midpoint type inequalities for the convex function, the function P, s-convex functions, and polynomial n-fractional convex functions, respectively, involving the Riemann-Liouville fractional operator, the Hadamard fractional operator, and the Katugompola fractional operators, respectively.
Theorem 6.2
Let \(\ p>1\), \(\frac{1}{p^{\prime }}+\frac{1}{p}=1\) and h be a B-function. Assume that \(\beta ,\psi \) are defined as in Lemma 6.1. If \(\left| f^{\prime }\right| ^{p}\) is a h-convex mapping on [a, b], we get the following midpoint type inequality
Proof
Apply Lemma 5.1 and the absolute value, we get
Using Hölder inequality and \(\displaystyle A^{\frac{1}{p}}+B^{\frac{1}{p} }\le 2^{1-\frac{1}{p}}(A+B)^{\frac{1}{p}}\), we get
Given that \(\left| f^{\prime }\right| ^{p}\) is a h-convex and h is B-function, we obtain
This accomplishes the first inequality in (6.14).
Notice that for \(p>1\) and \(A,B\ge 0\), we have \(A^{p}+B^{p}\le (A+B)^{p}\), which yields to the second inequality in (6.14). \(\square \)
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BB wrote first and second sections. NA wrote other sections. HB revised the paper.
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Benaissa, B., Azzouz, N. & Budak, H. Hermite-Hadamard type inequalities for new conditions on h-convex functions via \(\psi \)-Hilfer integral operators. Anal.Math.Phys. 14, 35 (2024). https://doi.org/10.1007/s13324-024-00893-3
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DOI: https://doi.org/10.1007/s13324-024-00893-3