Erratum | Open Access

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

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

- 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 - B.-Y. Xi, R.-F. Bai, and F. Qi, “Hermite-Hadamard type inequalities for the $m$- and $(\alpha ,m)$-geometrically convex functions,”
*Aequationes Mathematicae*, vol. 84, no. 3, pp. 261–269, 2012. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

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.