Keywords: HermiteHadamard type inequality, Hölder’s integral inequality, quasigeometrically convex function
American Journal of Mathematical Analysis, 2013 1 (3),
pp 4852.
DOI: 10.12691/ajma135
Received November 05, 2013; Revised November 22, 2013; Accepted December 10, 2013
Copyright © 2014 Science and Education Publishing. All Rights Reserved.
1. Introduction
Let I be on interval in . It is general knowledge that if is a convex function and a,b∈I with a<b, then
This inequality is well known in the literature as HermiteHadamard’s inequality for convex functions.
In ^{[4, 5]}, Niculescu defined the GAconvex and GGconvex functions as follows:
Definition 1.1. The function is said to be geometric arithmetically convex, simply speaking, GAconvex on I if the following inequality
holds for all and , where and are respectively called the weighted geometric mean of two positive numbers andand the weighted arithmetic mean of and .
Definition 1.2. The function is said to be geometrically convex, simply speaking, GGconvex on I if the following inequality
holds for all and .
For some recent results concerning HermiteHadamard type inequalities for GAconvex and GGconvex functions we refer interestes reader to ^{[1, 2, 3, 6, 7]}.
In ^{[2]}, Iscan defined the quasigeometricallyconvex functions as follows:
Definition 1.3. A function is said to be quasigeometrically convex on I if
For any and .
Clearly, any GAconvex and GGconvex functions are quasigeometrically convex functions. Furthermore, there exist quasigeometrically convex functions which are neither GAconvex nor GGconvex ^{[2]}.
In ^{[6]}, Park obtained generalized HermiteHadamard type integral inequalities for GAconvex functions by using the following lemma:
Lemma 1.1. Let be a diﬀerentiable function on , the interior of an interval and with . If then the following identity holds:
 (1.1) 
for n≠0.
In ^{[7]}, Zhang et al. established some HermiteHadamard type integral inequalities for GAconvex functions by using inequality (1.1) for . In ^{[3]} Ji et al. established some HermiteHadamard type integral inequalities for GAconvex functions by using inequality (1.1) for .
The goal of this article is to establish some new generalized integral inequalities of HermiteHadamard type for quasigeometrically convex functions by using inequality (1.1) and Hölder inequality.
Throughout this paper we will consider
the logarithmic mean for b>a>0.
2. HermiteHadamard Type Inequalities for Quasigeometrically Convex Functions
Theorem 2.1. Let be a diﬀerentiable function on , with and . If is quasigeometrically convex on for , then
for .
Proof. Since is quasigeometrically convex on , from Lemma 1.1 and Hölder’s inequality we have
Theorem 2.2. Let be a diﬀerentiable function on , with and . If is quasigeometrically convex on for , then
for .
Proof. Since is quasigeometrically convex on , from Lemma 1.1 and Hölder’s inequality we have
Theorem 2.3. Let be a diﬀerentiable function on , with and . If is quasigeometrically convex on for , then
for .
Proof. Since is quasigeometrically convex on , from Lemma 1.1 and Hölder’s inequality we have
Theorem 2.4. Let be a diﬀerentiable function on , with and . If is quasigeometrically convex on for with , then
for .
Proof. Since is quasigeometrically convex on , from Lemma 1.1 and Hölder’s inequality we have
Theorem 2.5. Let be a diﬀerentiable function on , with and . If is quasigeometrically convex on for with , then
for .
Proof. Since is quasigeometrically convex on , from Lemma 1.1 and Hölder’s inequality we have
Theorem 2.6. Let be a diﬀerentiable function on , with and . If is quasigeometrically convex on for with , then
for .
Proof. Since is quasigeometrically convex on , from Lemma 1.1 and Hölder’s inequality we have
Theorem 2.7. Let be a diﬀerentiable function on , with and . If is quasigeometrically convex on for with , then
for and .
Proof. Since is quasigeometrically convex on , from Lemma 1.1 and Hölder’s inequality we have
Theorem 2.8. Let be a diﬀerentiable function on , with and . If is quasigeometrically convex on for with , then
for and .
Proof. Since is quasigeometrically convex on , from Lemma 1.1 and Hölder’s inequality we have
Theorem 2.9. Let be a diﬀerentiable function on , with and . If is quasigeometrically convex on for with , then
for and .
Proof. Since is quasigeometrically convex on , from Lemma 1.1 and Hölder’s inequality we have
Theorem 2.10. Let be a diﬀerentiable function on , with and . If is quasigeometrically convex on for with , then
for and .
Proof. Since is quasigeometrically convex on , from Lemma 1.1 and Hölder’s inequality we have
Theorem 2.11. Let be a diﬀerentiable function on , with and . If is quasigeometrically convex on for with , then
for .
Proof. Since is quasigeometrically convex on , from Lemma 1.1 and Hölder’s inequality we have
3. Applications to Special Means
Finally we apply HermiteHadamard type inequalities obtained in the above section to construct some inequalities for special means.
Proposition 3.1. For , and , we have
where and
Proof. Let for , and . Then the function is quasigeometrically convex function on . Thus by the inequalities in Theorem 2.1, Theorem 2.2 and Theorem 2.3, Proposition 3.1 is proved.
Proposition 3.2. For , and with , we have
where ,and
Proof. Let for , and . Then the function is quasigeometrically convex function on . Thus by the inequalities in Theorem 2.4, Theorem 2.5, Theorem 2.6, Theorem 2.7, Theorem 2.8, Theorem 2.9, Theorem 2.10 and Theorem 2.11, Proposition 3.2 is proved.
References
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