Abstract and Applied Analysis

Abstract and Applied Analysis / 2014 / Article

Erratum | Open Access

Volume 2014 |Article ID 294739 | 5 pages | https://doi.org/10.1155/2014/294739

Erratum to “On Integral Inequalities of Hermite-Hadamard Type for s-Geometrically Convex Functions”

Received16 Jun 2013
Accepted13 Nov 2013
Published25 Mar 2014

In [1, Definition 1.9], the concept “s-geometrically convex function” was introduced.

Making use of [1, Lemma 2.1], Hölder’s integral inequality, and other analytic techniques, some inequalities of Hermite-Hadamard type were established. However, there are some vital errors appeared in main results of the paper [1].

The aim of this paper is to correct these errors and we now start off to correct them.

Correction to Theorem  3.1. Let be a differentiable function on such that for . If is s-geometrically convex and monotonically decreasing on for and , then where

Proof. Since is s-geometrically convex and monotonically decreasing on , using Lemma  2.1 and Hölder’s inequality gives
Let and . Then
When , we have When , by (7), we obtain When , by (7), we have Substituting (8) to (13) into (6) yields inequality (1).
Since is s-geometrically convex and monotonically decreasing on , by Lemma 2.1 and Hölder’s inequality, we obtain When , we have When , by (7), we obtain When , by (7), we have Substituting (15) to (20) into (14) leads to inequality (2). Theorem 3.1 is thus proved.

Correction to Corollary   3.2. Under the conditions of Theorem 3.1,(1)when , (2)when ,

Correction to Theorem  3.3. Let be a differentiable function on such that for . If is s-geometrically convex and monotonically decreasing on for and , then where is the same as in (4),

Proof. Since is s-geometrically convex and monotonically decreasing on , by Lemma 2.1 and Hölder’s inequality, we have
When , we have When , by (7), we obtain When , by (7), we have Substituting (28) to (33) into (26) and (27) results in inequalities (23) and (24). Theorem 3.3 is thus proved.

Correction to Corollary  3.4. Under the conditions of Theorem 3.3, when , we have

Correction to Theorem  4.1. Let , , and . Then where for and with are the arithmetic, logarithmic, and generalized logarithmic means, respectively.

If , then

Proof. Let , , and for . Then the function is s-geometrically convex on for , and . Therefore, By Theorem 3.1, Theorem   4.1 is thus proved.

Correction to Theorem  4.2. Let , , and . Then

Proof. It is easy to see that
Hence, by Theorem 3.3, Theorem 4.2 is thus proved.

Remark. By the way, all the powers which appeared four times in [2, Theorem 4.2 and Corollary  4.2] should be corrected as , respectively.

Acknowledgments

The authors would like to thank Professor Feng Qi in China for his valuable contributions to these corrections. This work was supported in part by the NNSF of China under Grant no. 11361038 and by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant no. NJZY14191 and NJZY13159, China.

References

  1. T.-Y. Zhang, A.-P. Ji, and F. Qi, “On integral inequalities of Hermite-Hadamard type for s-geometrically convex functions,” Abstract and Applied Analysis, vol. 2012, Article ID 560586, 14 pages, 2012. View at: Publisher Site | Google Scholar
  2. B.-Y. Xi, R.-F. Bai, and F. Qi, “Hermite-Hadamard type inequalities for the m- and (α,m)-geometrically convex functions,” Aequationes Mathematicae, vol. 84, no. 3, pp. 261–269, 2012. View at: Publisher Site | Google Scholar | MathSciNet

Copyright © 2014 Tian-Yu Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

725 Views | 440 Downloads | 2 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.