Keywords: integral identity, integral inequality, HermiteHadamard type, strongly quasiconvex function, Hölder inequality
Received July 07, 2016; Revised September 15, 2016; Accepted September 23, 2016
Copyright © 2016 Science and Education Publishing. All Rights Reserved.
1. Introduction
We first list some definitions concerning various convex functions.
Definition 1.1. A function is said to be convex if
 (1.1) 
holds for alland.
Deﬁnition 1.2 (^{[1]}). A function is said to be quasiconvex convex if
 (1.2) 
for all
Deﬁnition 1.3 (^{[2]}). A function is said to be strongly convex with modulus if
 (1.3) 
is valid for all and .
The following inequalities of the HermiteHadamard type were established for the above convex functions.
Theorem 1.1 (^{[3]}). Let be differentiable on and with .
(i) If is convex function on , then
 (1.4) 
(ii) If is convex function on and , then
 (1.5) 
In ^{[4]}, two inequalities of the HermiteHadamard type for quasiconvex functions were introduced as follow.
Theorem 1.2 (^{[4]}). Let be a differentiable mapping on and with . If is quasiconvex on , then
 (1.6) 
Theorem 1.3 (^{[4]}). Let be a differentiable mapping on and with . If is quasiconvex on and , then
 (1.7) 
Theorem 1.4 ([5, Theorem 2.3]). Let be a differentiable mapping on and with . If is quasiconvex on and , then
 (1.8) 
Theorem 1.5 ([5, Theorem2.4]). Let be a differentiable mapping on and with . If is quasiconvex on and , then
 (1.9) 
In this paper, we will introduce a new notion “strongly quasiconvex function” and establish some integral inequalities of the HermiteHadamard type for functions whose derivatives are of strongly quasiconvexity.
2. A Definition and a Lemma
We now introduce the notion “strongly quasiconvex functions”.
Deﬁnition 2.1. A function is said to be strongly quasiconvex with modulus if
is valid for all and .
For establishing new integral inequalities of the HermiteHadamard type for strongly quasiconvex functions, we need the following identity.
Lemma 2.1. Let be differentiable on and with . If , then
 (2.1) 
Proof. This follows from a straightforward computation of definite integrals.
3. Some Integral Inequalities of the HermiteHadamard Type
Now we are in a position to establish some integral inequalities of the Hermite–Hadamard type for functions whose derivatives are of strongly quasiconvexity.
Theorem 3.1. Let be differentiable mapping on and with . If and is strongly quasiconvex on for with modulus , then
Proof. Since is strongly quasiconvex on , using Lemma 2.1 and by Hölder’s inequality, we have
Theorem 3.1 is thus proved.
Crollary 3.1. Under conditions of Theorem 3.1, if , then
Theorem 3.2. Let be differentiable mapping on and with . If and is strongly quasiconvex on for with modulus , then
Proof. By Lemma 2.1 and using Hölder’s inequality and the strongly quasiconvexity of , we obtain
Theorem 3.3 is proved.
Acknowledgement
This work was partially supported by the National Natural Science Foundation under Grant No. 11361038 of and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant No. 2015MS0123, .
The authors thank the anonymous referee for his/her careful corrections to and valuable comments on the original version of this paper.
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