Page 161 - IJOCTA-15-1

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.1, pp.155-165 (2025)
https://doi.org/10.36922/ijocta.1568

RESEARCH ARTICLE

Improvements of Hermite-Hadamard-Mercer inequality using
k-fractional integral

*
Jamroz Khan, Muhammad Adil Khan , Sana Sarwar
Department of Mathematics,University of Peshawar, Peshawar, Pakistan
jamroz.khan73@gmail.com, madilkhan@uop.edu.pk, sanasarwar982@gmail.com

ARTICLE INFO ABSTRACT
Article History: The well-known Hermite-Hadamard inequality has attracted the attention of
Received: 22 March 2024 several researchers due to the fact that Hermite-Hadamard inequality has many
Accepted: 26 September 2024 important applications in mathematics as well as in other areas of science.
Available Online: 28 January 2025 In this article, the authors present new Hermite-Hadamard inequality of the
Mercer type containing Riemann-Liouville k-fractional integrals. For these
Keywords:
inequalities, we give integral identity for differentiable functions. With the
Hermite-Hadamard inequality
help of the identity and Hermite-Hadamard-Mercer type inequalities, we derive
Jensen-Mercer inequality
Riemann-Liouville fractional integrals several results for the inequalities. We establish bounds for the difference of the
k−fractional integral obtain results by applying H¨older’s inequality and power-mean inequality. We
hope that the proposed result will invigorate further interest in this direction.
AMS Classification 2010:
26D15, 26A33, 26A51

1. Introduction For example, the local minima of a convex func-
tion is also global minima of the function. Also
Convexity is one of the fundamental, powerful every convex function always produce a convex
and important notion in mathematical analysis set (i.e. epi graph of convex function). The con-
presented over 100 years ago. A function ω : vex functions and convex sets play pivotal role
[δ 1 , δ 2 ] → R is said to be convex (or concave up) in optimization theory and nonlinear analysis 2–5
if as well as in information theory. The idea of
6
convex functions is essential for setting bound-
aries. The convex functions have been general-
ω(tu + (1 − t)v) ≤ tω(u) + (1 − t)ω(v) (1) 7
ized in different way such as s-convex, Harmon-
ically convex, 8 h-convex function, 9 exponential
convexity, 10 m−convex functions, 11,12 4-convex
for all u, v ∈ [δ 1 , δ 2 ] and t ∈ [0, 1]. When the in- 13 15 14
functions, Raina’s function, P-convex and
equality in (1) is reversed then ω is called a con- coordinate convex 16 etc. Many important in-
cave function (or concave down). 1 equalities like Hermite-Hadamard’s (H-H) in-
The convex functions has applications in both equality, Jensen’s inequality, majorization in-
pure and applied mathematics. It has also ap- equality, Slater’s inequality, and Sherman’s in-
plications in other fields like economics, indus- equality are the outcome of convex functions.
try, business, engineering, nonlinear program- Many fields of science are generated via mathe-
ming, optimization theory and medicines etc. The matical inequalities. The mathematical inequal-
convex function has many important and inter- ities and convex functions are closely related to
esting properties that attract the researchers and each others. In recent years, the theory of in-
due to these properties convex functions play a equalities has progressed very fast and the theory
vital role in many areas of science.
*Corresponding Author
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