Research Article
Fractional Integral Inequalities of Hermite–Hadamard Type for
Differentiable Generalized h-Convex Functions
Yingxia Yang,
1
Muhammad Shoaib Saleem ,
2
Mamoona Ghafoor,
2
and Muhammad Imran Qureshi
3
1
School of Mathematics, Yunnan Normal University, Kunming 650500, China
2
Department of Mathematics, University of Okara, Okara 56300, Pakistan
3
Department of Mathematics, COMSATS University Islamabad, Vehari 61100, Pakistan
Correspondence should be addressed to Muhammad Shoaib Saleem; shaby455@yahoo.com
Received 17 March 2020; Accepted 19 May 2020; Published 25 June 2020
Academic Editor: Ming-Sheng Liu
Copyright©2020YingxiaYangetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In the present paper, some fractional integral inequalities of Hermite–Hadamard type for functions whose derivatives are
generalized h-convexareestablished.Moreover,severalparticularcasesarealsodiscussedwhichcanbededucedfromourresults.
Asspecialcases,onecanobtainseveralnewversionsoftheresultsofgeneralized h-convexityforothervariousgeneralizationsof
convex functions.
1.Introduction
e theory of inequalities is in process of continuous de-
velopmentstate,andinequalitieshavebecomeveryeffectual
andpotenttoolsforanalyzingalargenumberofproblemsin
different branches of mathematics. In last decades, the
theoryofinequalitieshasdrawntheattentionofmostofthe
researchers,motivatednewresearchdirections,andaffected
several features of mathematical analysis and applications
[1–4]. Among a variety of inequalities, some inequalities
such as Jensen, Hilbert, Hadamard, Hardy, Poincare,
Sobolev, Opial, and Fej´ er have made a significant influence
on numerous branches of mathematics [5–8]. One good
strategy for investigating any convexity is first to discuss
maininequalitiessuchasHermite–HadamardandFej´ erand
then to derive fractional integral inequalities for this. e
fractionalinequalitiesforconvexfunctionarealsoimportant
to calculate different means. So, it is always interesting to
develop fractional integral inequalities for some generalized
convexity. For recent results concerning fractional Hermi-
te–Hadamard inequalities and for different generalizations
of convexity, see [9–11] and references therein.
In 2019, some new inequalities based on harmonic log-
convex functions have been studied by Baloch et al. in [12].
In the same year, Sarikaya and Alp [13] studied the
Hermite–Hadamard–Fej´ er type integral inequalities for
generalized convex functions via local fractional integrals.
Kwun et al. [14] studied generalized Riemann–Liouville
k-fractional integrals associated with Ostrowski-type in-
equalities and error bounds of Hadamard inequalities. In
[15], Kang et al. studied Hadamard and Fej´ er–Hadamard
inequalities for extended generalized fractional integrals
involvingspecialfunctions.In[15], (h, m)-convexfunctions
and associated fractional Hadamard and Fej´ er–Hadamard
inequalities via an extended generalized Mittag–Leffler
function were studied. Simpson’s type inequalities for
strongly (s, m)− convex functions in the second sense has
been studied in [16] by Kermausuor.
In the present paper, we aim to study some Hermi-
te–Hadamard-type fractional integral inequalities for
functions whose derivatives are generalized h-convex. Our
results can be considered as generalization of many existing
resultsasmanyexistingresultscanbeobtaineddirectlyfrom
our results.
Hindawi
Journal of Mathematics
Volume 2020, Article ID 2301606, 13 pages
https://doi.org/10.1155/2020/2301606