Abstract
In the paper, the authors establish some new Hermite–Hadamard type inequalities for harmonically convex functions via generalized fractional integrals. Moreover, the authors prove extensions of the Hermite–Hadamard inequality for harmonically convex functions via generalized fractional integrals without using the harmonic convexity property for the functions. The results offered here are the refinements of the existing results for harmonically convex functions.
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1 Introduction
The Hermite–Hadamard inequality, which is the first basic result of convex mappings with a natural geometric interpretation and extensive use, has attracted attention with great interest in elementary mathematics. Many mathematicians have devoted their efforts to standardization, refining, imitation, and expansion into various categories of works such as convex mappings.
Inequalities found by C. Hermite and J. Hadamard for convex mappings are very important in literature (see [1]). These inequalities state that if \(\mathcal{F}:I\rightarrow \mathbb{R}\) is a convex function on the interval I of real numbers and \(\kappa _{1},\kappa _{2}\in I\) with \(\kappa _{1}<\kappa _{2}\), then
Both inequalities hold in the reversed direction if \(\mathcal{F}\) is concave. For further studies of this area, one can consult [2–22].
For brevity, in the upcoming results, we use the subsequent notations: Mappings \(\Lambda,\Lambda ^{\ast },\Psi,\Psi ^{\ast }: [ 0,1 ] \rightarrow \mathbb{R} \) are defined by
and
Now we give the definition of the generalized fractional integrals (GFIs) given by Sarikaya and Ertuğral in [23].
Definition 1
The left-sided and right-sided GFIs are denoted by \({}_{\kappa _{1}+}I_{\varphi }\) and \({}_{\kappa _{2}-}I_{\varphi }\) and defined as follows:
where a function \(\varphi:[0,\infty )\rightarrow {}[ 0,\infty )\) satisfies the condition \(\int _{0}^{1}\frac{\varphi ( \tau ) }{\tau }\,d\tau <\infty \).
Recently, the authors gave some refinements of Hermite–Hadamard inequalities for GFIs under the condition of convexity, as follows.
Theorem 1
([23])
For a convex function \(\mathcal{F}: [ \kappa _{1},\kappa _{2} ] \rightarrow \mathbb{R} \) on \([ \kappa _{1},\kappa _{2} ] \) with \(\kappa _{1}<\kappa _{2}\), the subsequent inequalities hold for GFIs:
Theorem 2
([24])
For a convex function \(\mathcal{F}: [ \kappa _{1},\kappa _{2} ] \rightarrow \mathbb{R} \) on \([ \kappa _{1},\kappa _{2} ] \) with \(\kappa _{1}<\kappa _{2}\), the subsequent inequalities hold for GFIs:
Theorem 3
([25])
For a convex function \(\mathcal{F}: [ \kappa _{1},\kappa _{2} ] \rightarrow \mathbb{R} \) on \([ \kappa _{1},\kappa _{2} ] \) with \(\kappa _{1}<\kappa _{2}\), the subsequent inequalities hold for GFIs:
In [26], İşcan gave the following definition of harmonically convex functions and Hermite–Hadamard inequalities for harmonically convex functions.
Definition 2
([26])
A mapping \(\mathcal{F}:I\subseteq \mathbb{R} \backslash \{0\}\rightarrow \mathbb{R} \) is called harmonically convex if
for all \(\kappa _{1},\kappa _{2}\) in I and τ in \([0,1]\). If inequality (1.7) holds in the reversed direction, then \(\mathcal{F}\) is called a harmonically concave function.
Theorem 4
([26])
For a harmonically convex mapping \(\mathcal{F}:I\subseteq \mathbb{R} \backslash \{0\}\rightarrow \mathbb{R} \), the following double inequality holds:
where \(\kappa _{1},\kappa _{2}\in I\) and \(\kappa _{1}<\kappa _{2}\).
In [27], İşcan and Wu gave the inequalities of Hermite–Hadamard type for harmonically convex functions via Riemann–Liouville fractional integrals.
Theorem 5
([27])
Let \(\mathcal{F}:I\subseteq (0,\infty )\rightarrow \mathbb{R} \) be a function such that \(\mathcal{F}\in L([\kappa _{1},\kappa _{2}])\), where \(\kappa _{1},\kappa _{2}\in I\) with \(\kappa _{1}<\kappa _{2}\). If \(\mathcal{F}\) is a harmonically convex function on \([\kappa _{1},\kappa _{2}]\), the following double inequality holds for the Riemann–Liouville fractional integrals:
where \(\alpha > 0\).
In [28], Zhao et al. gave the following Hermite–Hadamard type inequalities for harmonically convex functions by utilizing GFIs.
Theorem 6
Let \(\mathcal{F}:I\subseteq (0,+\infty )\rightarrow \mathbb{R} \) be a mapping such that \(\mathcal{F}\in L([\kappa _{1},\kappa _{2}])\). If \(\mathcal{F}\) is a harmonically convex mapping on \([\kappa _{1},\kappa _{2}]\), then the following inequalities hold for the GFIs:
In [29], F. Chen gave the following useful lemma and the lower and upper bounds of the left- and right-hand sides of inequalities (1.9) as follows.
Lemma 1
A mapping \(\mathcal{F}:[\kappa _{1},\kappa _{2}]\subseteq \mathbb{R} \backslash \{0\}\rightarrow \mathbb{R} \) is called harmonically convex if and only if \(\phi ( x ) \) is convex on \([\kappa _{1},\kappa _{2}]\).
Theorem 7
Let \(\mathcal{F}: [ \kappa _{1},\kappa _{2} ] \subseteq ( 0,+\infty ) \rightarrow \mathbb{R} \) be a positive twice differentiable function with \(\kappa _{1}<\kappa _{2}\) and \(\mathcal{F}\in L ( [ \kappa _{1},\kappa _{2} ] )\).If \(\phi ^{\prime \prime }\) is bounded in \([ \kappa _{1},\kappa _{2} ] \), then the following inequalities hold for the Riemann–Liouville fractional integrals:
Theorem 8
Let \(\mathcal{F}: [ \kappa _{1},\kappa _{2} ] \subseteq ( 0,+\infty ) \rightarrow \mathbb{R} \) be a positive twice differentiable function with \(\kappa _{1}<\kappa _{2}\) and \(\mathcal{F}\in L ( [ \kappa _{1},\kappa _{2} ] )\).If \(\phi ^{\prime \prime }\) is bounded in \([ \kappa _{1},\kappa _{2} ] \), then the following inequalities hold for the Riemann–Liouville fractional integrals:
In [30], Budak et al. gave the following inequalities for harmonically convex mappings.
Theorem 9
Let \(\mathcal{F}: [ \kappa _{1},\kappa _{2} ] \subseteq ( 0,+\infty ) \rightarrow \mathbb{R} \) be a positive twice differentiable function with \(\kappa _{1}<\kappa _{2}\) and \(\mathcal{F}\in L ( [ \kappa _{1},\kappa _{2} ] )\).If \(\phi ^{\prime \prime }\) is bounded in \([ \kappa _{1},\kappa _{2} ] \), then the following inequalities hold for GFIs:
and
For recent findings and implications for integral inequalities via harmonically convex mappings and other classes of functions, see ([31–42]) and the references given therein.
2 Hermite–Hadamard type inequalities
In this portion, we deal with some new inequalities of Hermite–Hadamard type for harmonically convex mappings by applying GFIs.
Theorem 10
Let \(\mathcal{F}:I\subseteq (0,+\infty )\rightarrow \mathbb{R} \) be a function such that \(\mathcal{F}\in L([\kappa _{1},\kappa _{2}])\). If \(\mathcal{F}\) is a harmonically convex function on \([\kappa _{1},\kappa _{2}]\), then the following inequalities hold for the GFIs:
Proof
From harmonic convexity, we have
For \(x= \frac{2\kappa _{1}\kappa _{2}}{\tau \kappa _{1}+ ( 2-\tau ) \kappa _{2}}\) and \(y= \frac{2\kappa _{1}\kappa _{2}}{ ( 2-\tau ) \kappa _{1}+\tau \kappa _{2}}\), we get
Multiplying by \(\frac{\varphi ( \frac{(\kappa _{2}-\kappa _{1})}{2\kappa _{1}\kappa _{2}}\tau ) }{\tau }\) both sides of inequality (2.2) and integrating the resultant one with respect to τ over \([0,1]\), we obtain
For \(\frac{1}{u}= \frac{2\kappa _{1}\kappa _{2}}{\tau \kappa _{1}+ ( 2-\tau ) \kappa _{2}}\) and \(\frac{1}{v}= \frac{2\kappa _{1}\kappa _{2}}{ ( 2-\tau ) \kappa _{1}+\tau \kappa _{2}}\), we obtain
Hence, we proved the first inequality. To prove the second inequality of (2.1), first we note that since \(\mathcal{F}\) is a harmonically convex function, we have
and
Adding (2.4) and (2.5), we get
Multiplying by \(\frac{\varphi ( \frac{(\kappa _{2}-\kappa _{1})}{2\kappa _{1}\kappa _{2}}\tau ) }{\tau }\) both sides of inequality (2.6) and integrating the resultant one with respect to τ over \([0,1]\), we obtain
By changing the variables of integration, we have the second inequality of (2.1). □
Remark 1
Under the assumptions of Theorem 10, if we put \(\varphi ( \tau ) =\tau \), then Theorem 10 reduces to Theorem 4.
Corollary 1
Under the assumptions of Theorem 10, if we set \(\varphi ( \tau ) = \frac{\tau ^{\alpha }}{\Gamma ( \alpha ) }\), then we have the following inequality for Riemann–Liouville fractional integrals:
Corollary 2
Under the assumptions of Theorem 10, if we set \(\varphi ( \tau ) = \frac{\tau ^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \alpha ) }\), then we have the following inequality for the k-Riemann–Liouville fractional integrals:
Theorem 11
Let \(\mathcal{F}:I\subseteq (0,+\infty )\rightarrow \mathbb{R} \) be a function such that \(\mathcal{F}\in L([\kappa _{1},\kappa _{2}])\). If \(\mathcal{F}\) is a harmonically convex function on \([\kappa _{1},\kappa _{2}]\), then the following inequalities hold for the GFIs:
Proof
Since \(\mathcal{F}\) is a harmonically convex function on \([\kappa _{1},\kappa _{2}]\), we have
For \(x= \frac{2\kappa _{1}\kappa _{2}}{ ( 1-\tau ) \kappa _{1}+ ( 1+\tau ) \kappa _{2}}\) and \(y= \frac{2\kappa _{1}\kappa _{2}}{ ( 1+\tau ) \kappa _{1}+ ( 1-\tau ) \kappa _{2}}\), we get
Multiplying by \(\frac{\varphi ( \frac{(\kappa _{2}-\kappa _{1})}{2\kappa _{1}\kappa _{2}}\tau ) }{\tau }\) both sides of inequality (2.8) and integrating the resultant one with respect to τ over \([0,1]\), we obtain
By setting \(\frac{1}{u}= \frac{2\kappa _{1}\kappa _{2}}{ ( 1-\tau ) \kappa _{1}+ ( 1+\tau ) \kappa _{2}}\) and \(\frac{1}{v}= \frac{2\kappa _{1}\kappa _{2}}{ ( 1+\tau ) \kappa _{1}+ ( 1-\tau ) \kappa _{2}}\), we have
Hence we have the first inequality in (2.7).
To prove the second inequality in (2.7), first we note that \(\mathcal{F} \) is a harmonically convex function, we get
and
Adding (2.9) and (2.10), we have
Multiplying by \(\frac{\varphi ( \frac{(\kappa _{2}-\kappa _{1})}{2\kappa _{1}\kappa _{2}}\tau ) }{\tau }\) both sides of inequality (2.11) and integrating the resultant one with respect to τ over \([0,1]\), we obtain
By changing the variable of integration, we have the second inequality in (2.7). □
Remark 2
Under the assumptions of Theorem 11, if we put \(\varphi ( \tau ) =\tau \), then Theorem 10 reduces to Theorem 4.
Corollary 3
Under the assumptions of Theorem 11, if we set \(\varphi ( \tau ) = \frac{\tau ^{\alpha }}{\Gamma ( \alpha ) }\), then we have the following inequalities for the Riemann–Liouville fractional integrals:
Corollary 4
Under the assumptions of Theorem 11, if we put \(\varphi ( \tau ) = \frac{\tau ^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \alpha ) }\), then we have the following inequalities for the k-Riemann–Liouville fractional integrals:
3 Extension of Hermite–Hadamard type inequalities
In this section, we give the following inequalities which give the above and below bounds for the left- and right-hand sides of inequalities (2.1) and (2.7). We prove inequalities (2.1) and (2.7) under the condition \(\phi ^{\prime } ( \kappa _{1}+\kappa _{2}-x ) \geq \phi ^{ \prime }(x)\) instead of the harmonic convexity of \(\mathcal{F}\).
Theorem 12
Let \(\mathcal{F}: [ \kappa _{1},\kappa _{2} ] \subseteq ( 0,+\infty ) \rightarrow \mathbb{R} \) be a positive twice differentiable function with \(\kappa _{1}<\kappa _{2}\) and \(\mathcal{F}\in L ( [ \kappa _{1},\kappa _{2} ] ) \). If \(\phi ^{\prime \prime }\) is bounded in \([ \kappa _{1},\kappa _{2} ] \), then the following inequalities hold for GFIs:
Proof
By using the change of variables, we have
By equality (3.2), we get
Using the fact that
and
we have
We also have
By using equality (3.5) and the assumption \(m<\phi ^{\prime \prime }(u)<M\), \(u\in [ \kappa _{1},\kappa _{2} ] \), we obtain
i.e.
Integrating inequality (3.6) with respect to τ on \([ x,\frac{\kappa _{1}+\kappa _{2}}{2} ] \), we get
By equality (3.4), we have
Multiplying inequality (3.7) by \(\frac{\varphi ( \frac{x-\kappa _{1}}{\kappa _{1}\kappa _{2}} ) }{2\Psi (1) ( x-\kappa _{1} ) } \) and integrating the resultant inequality with respect to x on \([ \kappa _{1},\frac{\kappa _{1}+\kappa _{2}}{2} ] \), we establish
That is, we get
which gives inequality (3.1). □
Remark 3
Under the assumptions of Theorem 12, if we put \(\varphi ( \tau ) =\tau \), then we have the following inequalities:
Corollary 5
Under the assumptions of Theorem 12, if we set \(\varphi ( \tau ) = \frac{\tau ^{\alpha }}{\Gamma ( \alpha ) }\), then we have the following inequalities for the Riemann–Liouville fractional integrals:
Corollary 6
Under the assumptions of Theorem 12, if we put \(\varphi ( \tau ) = \frac{\tau ^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \alpha ) }\), then we have the following inequalities for the k-Riemann–Liouville fractional integrals:
Theorem 13
Let \(\mathcal{F}: [ \kappa _{1},\kappa _{2} ] \subseteq ( 0,+\infty ) \rightarrow \mathbb{R} \) be a positive twice differentiable function with \(\kappa _{1}<\kappa _{2}\) and \(\mathcal{F}\in L ( [ \kappa _{1},\kappa _{2} ] ) \). If \(\phi ^{\prime \prime }\) is bounded in \([ \kappa _{1},\kappa _{2} ] \), then the following inequalities hold for the GFIs:
Proof
By using the change of variables, we have
By using the equalities
and
we have
Integrating (3.6) with respect to τ over \([ \kappa _{1},x ] \), we get
which implies that
Multiplying inequality (3.12) by \(\frac{\varphi ( \frac{x-\kappa _{1}}{\kappa _{1}\kappa _{2}} ) }{2\Psi (1) ( x-\kappa _{1} ) }\) and integrating the resultant inequality with respect to x on \([ \kappa _{1},\frac{\kappa _{1}+\kappa _{2}}{2} ] \), we establish
That can be written as
which gives inequalities (3.9). □
Remark 4
Under the assumptions of Theorem 13, if we put \(\varphi ( \tau ) =\tau \), then we have the following inequalities:
Corollary 7
Under the assumptions of Theorem 13, if we set \(\varphi ( \tau ) = \frac{\tau ^{\alpha }}{\Gamma ( \alpha ) }\), then we have the following inequalities for the Riemann–Liouville fractional integrals:
Corollary 8
Under the assumptions of Theorem 13, if we put \(\varphi ( \tau ) = \frac{\tau ^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \alpha ) }\), then we have the following inequalities for the k-Riemann–Liouville fractional integrals:
Now by using Theorems 12 and 13, we prove inequality (2.1) under the condition \(\phi ^{\prime } ( \kappa _{1}+\kappa _{2}-x ) \geq \phi ^{ \prime }(x)\) instead of the harmonic convexity of \(\mathcal{F}\).
Theorem 14
Let \(\mathcal{F}: [ \kappa _{1},\kappa _{2} ] \subseteq ( 0,+\infty ) \rightarrow \mathbb{R} \) be a positive twice differentiable function with \(\kappa _{1}<\kappa _{2}\) and \(\mathcal{F}\in L ( [ \kappa _{1},\kappa _{2} ] ) \). If \(\phi ^{\prime } ( \kappa _{1}+\kappa _{2}-x ) \geq \phi ^{ \prime }(x), \forall x\in [ \kappa _{1}, \frac{\kappa _{1}+\kappa _{2}}{2} ] \), then we have the following inequalities for GFIs:
Proof
which gives the first inequality in (3.14). On the other hand, by equalities (3.10) and (3.11), we have
This gives the second inequality in (3.14) and completes the proof. □
Theorem 15
Let \(\mathcal{F}: [ \kappa _{1},\kappa _{2} ] \subseteq ( 0,+\infty ) \rightarrow \mathbb{R} \) be a positive twice differentiable function with \(\kappa _{1}<\kappa _{2}\) and \(\mathcal{F}\in L ( [ \kappa _{1},\kappa _{2} ] ) \). If \(\phi ^{\prime \prime }\) is bounded in \([ \kappa _{1},\kappa _{2} ] \), then the following inequalities hold for the GFIs:
Proof
By using the change of variables, we have
By equality (3.16), we get
Using the fact that
and
we have
We also have
By using equality (3.19) and the assumption \(m<\phi ^{\prime \prime }(u)<M\), \(u\in [ \kappa _{1},\kappa _{2} ] \), we obtain
i.e.
Integrating inequality (3.20) with respect to τ on \([ x,\frac{\kappa _{1}+\kappa _{2}}{2} ] \), we get
By equality (3.18), we have
Multiplying inequality (3.21) by \(\frac{\varphi ( \frac{\kappa _{1}+\kappa _{2}}{2\kappa _{1}\kappa _{2}}-\frac{x}{\kappa _{1}\kappa _{2}} ) }{2\Psi (1) ( \frac{\kappa _{1}+\kappa _{2}}{2}-x ) }\) and integrating the resultant inequality with respect to x on \([ \kappa _{1},\frac{\kappa _{1}+\kappa _{2}}{2} ] \), we establish
That is, we get
which gives inequality (3.15). □
Remark 5
Under the assumptions of Theorem 15, if we put \(\varphi ( \tau ) =\tau \), then inequality (3.15) reduces to inequality (3.8).
Corollary 9
Under the assumptions of Theorem 15, if we set \(\varphi ( \tau ) = \frac{\tau ^{\alpha }}{\Gamma ( \alpha ) }\), then we have the following inequalities for the Riemann–Liouville fractional integrals:
Corollary 10
Under the assumptions of Theorem 15, if we put \(\varphi ( \tau ) = \frac{\tau ^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \alpha ) }\), then we have the following inequalities for the k-Riemann–Liouville fractional integrals:
Theorem 16
Let \(\mathcal{F}: [ \kappa _{1},\kappa _{2} ] \subseteq ( 0,+\infty ) \rightarrow \mathbb{R} \) be a positive twice differentiable function with \(\kappa _{1}<\kappa _{2}\) and \(\mathcal{F}\in L ( [ \kappa _{1},\kappa _{2} ] ) \). If \(\phi ^{\prime \prime }\) is bounded in \([ \kappa _{1},\kappa _{2} ] \), then the following inequalities hold for the GFIs:
Proof
By using the change of variables, we have
By using the equalities
and
we have
Integrating (3.6) with respect to τ over \([ \kappa _{1},x ] \), we get
which implies that
Multiplying inequality (3.25) by \(\frac{\varphi ( \frac{\kappa _{1}+\kappa _{2}}{2\kappa _{1}\kappa _{2}}-\frac{x}{\kappa _{1}\kappa _{2}} ) }{2\Psi ( 1 ) ( \frac{\kappa _{1}+\kappa _{2}}{2}-x ) }\) and integrating the resultant inequality with respect to x on \([ \kappa _{1},\frac{\kappa _{1}+\kappa _{2}}{2} ] \), we establish
That can be written as
which gives inequalities (3.22). □
Remark 6
Under the assumptions of Theorem 16, if we put \(\varphi ( \tau ) =\tau \), then inequality (3.22) reduces to inequality (3.13).
Corollary 11
Under the assumptions of Theorem 16, if we set \(\varphi ( \tau ) = \frac{\tau ^{\alpha }}{\Gamma ( \alpha ) }\), then we have the following inequalities for the Riemann–Liouville fractional integrals:
Corollary 12
Under the assumptions of Theorem 16, if we put \(\varphi ( \tau ) = \frac{\tau ^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \alpha ) }\), then we have the following inequalities for the k-Riemann–Liouville fractional integrals:
Now, by using Theorems 15 and 16, we prove inequality (2.7) under the condition \(\phi ^{\prime } ( \kappa _{1}+\kappa _{2}-x ) \geq \phi ^{ \prime }(x)\) instead of the harmonic convexity of \(\mathcal{F}\).
Theorem 17
Let \(\mathcal{F}: [ \kappa _{1},\kappa _{2} ] \subseteq ( 0,+\infty ) \rightarrow \mathbb{R} \) be a positive twice differentiable function with \(\kappa _{1}<\kappa _{2}\) and \(\mathcal{F}\in L ( [ \kappa _{1},\kappa _{2} ] ) \). If \(\phi ^{\prime } ( \kappa _{1}+\kappa _{2}-x ) \geq \phi ^{ \prime }(x), \forall x\in [ \kappa _{1}, \frac{\kappa _{1}+\kappa _{2}}{2} ] \), then we have the following inequalities for GFIs:
Proof
From (3.17) and (3.18), we get
which proves the first inequality in (3.26). On the other hand, by equalities (3.23) and (3.24), we have
This proves the second inequality in (3.26) and completes the proof. □
4 Conclusion
In this work, the authors established Hermite–Hadamard type inequalities for harmonically convex functions by using generalized fractional integrals. Furthermore, the authors proved some extensions of newly proved inequalities without using the condition of harmonic convexity for the functions. It is an interesting and new problem, and the upcoming researchers can offer similar inequalities for harmonically convex functions on the co-ordinates via generalized fractional integrals in their future research.
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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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The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485, 11971241) and Philosophy and Social Sciences of Educational Commission of Hubei Province of China (Grant No. 20Y109).
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You, XX., Ali, M.A., Budak, H. et al. Extensions of Hermite–Hadamard inequalities for harmonically convex functions via generalized fractional integrals. J Inequal Appl 2021, 102 (2021). https://doi.org/10.1186/s13660-021-02638-3
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DOI: https://doi.org/10.1186/s13660-021-02638-3