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 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.