Page 162 - IJOCTA-15-1

J. Khan, M. Adil Khan, S. Sarwar / IJOCTA, Vol.15, No.1, pp.155-165 (2025)
of convex functions play influential role in the de- i = 1, 2, ..., n. Then the inequality
velopment of the inequalities theory. 17–22 n ! n
X X
ω δ 1 + δ 2 − q i u i ≤ ω(δ 1 )+ω(δ 2 )− q i ω(u i )
The H-H integral inequality was first notice by i=1 i=1
Hermite in 1883 23 and J. Hadamard rediscovered (3)
it independently ten years later. 24,25 It gives an holds.
estimates of the integral mean of any convex func-
The fractional calculus has important applica-
tion defined on an interval and is stated as: tions in mathematics, economics, engineering etc.
Let ω : [δ 1 , δ 2 ] → R be a convex function then the The fractional integral operators play a leading
H-H inequality is 26 role in fractional calculus. 36,37 Many important

inequalities has been refined, generalized and ex-
tended by using fractional integral operators. The

ω δ 1 +δ 2 ≤ 1 R δ 2 ω(u)du ≤ ω(δ 1 )+ω(δ 2 ) . Riemann-Liouville (RL) k−fractional integral op-
2 δ 2 −δ 1 δ 1 2
erators for a function ω on an interval [δ 1 , δ 2 ]
(2) are 38,39 :

Both the inequalities holds in the reverse direction Z x
1 γ
when ω is concave function. The equalities in (2) J + ω(u) = (u − x) k −1 ω(x)dx; u > δ 1 ,
γ,k
hold when the function ω is affine. The inequality δ 1 kΓ k (γ) δ 1
(2) has wide applications in mathematical analy- (4)
sis, engineering, economics etc. and is an area of J − ω(u) = 1 Z δ 2 (x − u) k γ −1 ω(x)dx; δ 2 > u
γ,k
great interest for the researchers. This inequality δ 2 kΓ k (γ) x
provide necessary and sufficient conditions for a (5)
function to be convex. The H-H inequality is in
elegant and simple form therefore it is a natural where γ is the order of ω and Γ k (γ) represents the
object of investigations. This inequality has rich k−gamma function, defined by
geometrical significance and some of the classical
Z ∞
k
inequalities for mean can be obtained for particu- Γ k (γ) = u γ−1 − u k du.
e
lar selection of the function ω. The H-H inequality 0
ensure the integrability of a convex function and γ,k γ,k
gives an estimate from both sides of the mean of a The operators J + ω(u), J − ω(u) given in (4), (5)
δ
δ
2
1
convex function. This inequality has attained the are called the left and right RL k-fractional inte-
attention in recent years and a lot of literature gral operators respectively. For k = 1, (4) and (5)
have been devoted to it providing new proofs, gen- reduce to left and right RL- fractional integral op-
eralizations, extensions, refinements and applica- erators respectively.
tions. In, 27–29 the conformable integral version of Sarikaya et. al. in, 40 use the idea of fractional cal-
H-H type inequalities have been obtained through culus for establishing Hermite-Hadamard inequal-
some integral identities. Iqbal et al, 30 have uti- ities containing fractional integrals. This result
lized the Green convex functions for the deriva- open a new path for researcher and is given as:
tion of the right Riemann-Liouville fractional in-
equalities. In, 31,32 the preinvex functions while Theorem 1. Suppose ω : [δ 1 , δ 2 ] → R is a posi-
in 33 coordinated convex functions played main tive convex function such that ω ∈ L [δ 1 , δ 2 ], then
role for the derivation of several H-H type inequal- the fractional integrals inequality
ities with their applications.
h i
The Jensen’s inequality is one the dynamic and ω δ 1 + δ 2 ≤ Γ(γ + 1) J + ω(δ 2 ) + J − ω(δ 1 )
γ
γ
fundamental inequality in the current literature 2 2 (δ 2 − δ 1 ) γ δ 1 δ 2
of convex functions and has wide applications in ω(δ 1 ) + ω(δ 2 )
various fields. Many classical inequalities such ≤ 2
as Minkowski’s inequality, Ky Fan’s inequality,
H¨older’s inequality, Young’s inequality, Levin- holds for all γ > 0.
son’s inequaliy, etc. can be obtain from Jensen’s
˙ I¸scan in, 41 gave the H-H inequality of Fej´er type
inequality. Mercer presented a variant of Jensen’s
inequality called Jensen-Mercer inequality. 34,35 that involve the left and right RL fractional inte-
gral operators.
This inequality stated as:
Let ω : [δ 1 , δ 2 ] → R be a convex function, Theorem 2. Suppose ω : [δ 1 , δ 2 ] → R is a
P n
q
u i ∈ [δ 1 , δ 2 ] and q i ∈ [0, 1] with i=1 i = 1 for convex function such that ω ∈ L [δ 1 , δ 2 ]. Let
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