Keywords: quasigeometrically convex functions, hermite–hadamard type inequalities, simpson type inequality
Turkish Journal of Analysis and Number Theory, 2014 2 (2),
pp 4246.
DOI: 10.12691/tjant223
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
Let real function be defined on some nonempty interval of real line R. The function is said to be convex on if inequality
holds for all and
Following inequalities are well known in the literature as HermiteHadamard inequality and Simpson inequality respectively:
Theorem 1. Let be a convex function defined on the interval of real numbers and with . The following double inequality holds
Theorem 2. Let be a four times continuously differentiable mapping on and Then the following inequality holds:
In recent years, many athors have studied errors estimations for HermiteHadamard, Ostrowski and Simpson inequalities; for refinements, counterparts, generalization see ^{[2, 9, 10]}.
The following definitions are well known in the literature.
Definition 1 (^{[7, 8]}). A function is said to be GAconvex (geometricarithmatically convex) if
for all and .
Definition 2 (^{[7, 8]}). A function is said to be GGconvex (called in ^{[13]} geometrically convex function) if
for all and .
In ^{[3]}, İşcan gave definition of quasigeometrically convexity as follows:
Definition 3. A function is said to be quasigeometrically convex on if
for any and
Clearly, any GAconvex and geometrically convex functions are quasigeometrically convex functions. Furthermore, there exist quasigeometrically convex functions which are neither GAconvex nor GGconvex ^{[3]}.
For some recent results concerning HermiteHadamard type inequalities for GAconvex, GGconvex, quasigeometrically convex functions we refer interestes reader to ^{[1, 3, 4, 5, 6, 11, 12, 14]}.
The goal of this article is to establish some new general integral inequalities of HermiteHadamard and Simpson type for quasigeometrically convex functions by using a new integral identity.
2. Main Results
Let be a differentiable function on , the interior of , throughout this section we will take
where with and .
In order to prove our main results we need the following identity.
Lemma 1. Let be a differentiable function on such that , where with Then for all we have:
 (1) 
Proof. By integration by parts and changing the variable, we can state
and similarly we get
Adding the resulting identities we obtain the desired result.
Theorem 3 Let be a differentiable function on such that , where with . If is quasigeometrically convex on for some fixed and , then the following inequality holds
 (2) 
where
 (3) 
and is logarithmic mean defined by
Proof. Since is quasigeometrically convex on for all
Hence, using Lemma 1 and power mean inequality we get
where
which completes the proof.
Corollary 1 Under the assumptions of Theorem 3 with the inequality (2) reduced to the following inequality
Corollary 2 Under the assumptions of Theorem 3 with and the inequality (2) reduced to the following inequality
Corollary 3 Under the assumptions of Theorem 3 with and the inequality (2) reduced to the following inequality
Theorem 4 Let be a differentiable function on such that , where with . If is quasigeometrically convex on for some fixed and , then the following inequality holds.
 (4) 
where
and .
Proof. Since is quasigeometrically convex on and using Lemma 1 and Hölder inequality, we get
here it is seen by simple computation that
Hence, the proof is completed.
Corollary 4 Under the assumptions of Theorem 4 with the inequality (4) reduced to the following inequality
Corollary 5 Under the assumptions of Theorem 4 with and the inequality (4) reduced to the following inequality.
Corollary 6 Under the assumptions of Theorem 4 with and the inequality (4) reduced to the following inequality
Theorem 5 Let be a differentiable function on such that , where with . If is quasigeometrically convex on for some fixed and , then the following inequality holds
 (5) 
where are defined as in Theorem 4 and .
Proof. Since is quasigeometrically convex on and using Lemma 1 and Hölder inequality, we get
Corollary 7 Under the assumptions of Theorem 5 with the inequality (5) reduced to the following inequality
Corollary 8 Under the assumptions of Theorem 5 with and the inequality (5) reduced to the following inequality
Corollary 9 Under the assumptions of Theorem 5 with and the inequality (5) reduced to the following inequality
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