Generalization of Stolarsky Type Means
© J. Pečarić and G. Roqia. 2010
Received: 27 April 2010
Accepted: 15 October 2010
Published: 20 October 2010
We generalize means of Stolarsky type and show the monotonicity of these generalized means.
1. Introduction and Preliminaries
This result for convex functions plays an important role in nonlinear analysis. These classical inequalities have been improved and generalized in a number of ways and applied for special means including Stolarsky type, logarithmic, and -logarithmic means. A generalization of H.H inequalities was obtained in [3–5], [2, page 5], and [1, page 143].
Some simple proofs of inequality (1.8) and related results on means of Stolarsky type are given in .
The aim of this paper is to prove the exponential convexity of the functions deduced from (1.5) and apply these functions to generalize the means of Stolarsky type, and at last we prove the monotonicity property of these new means.
We review some necessary definitions and preliminary results.
Definition 1.4 (see ).
Proposition 1.5 (see ).
Definition 1.6 (see ).
Corollary 1.7 (see ).
The following lemma is another way to define convex function [1, page 2].
In Section 2, we prove the exponential and logarithmic convexity of the functions deduced from (1.5). We also prove related mean value theorems of Cauchy type.
2. Main Results
The following lemma gives us very important family of convex functions.
Lemma 2.1 (see ).
(i)Consider the function
which implies (2.4).
We will use the following lemma in the proof of mean value theorem.
Lemma 2.3 (see [1, page 4]).
We get the required result.
which is clearly (2.20).
3. Means of Stolarsky Type
The proof of this Lemma is given in .
If we substitute and replace and in , for , then means of Stolarsky type and related results given in  are obtained.
4. Generalized Means of Stolarsky Type
The proof is similar to the proof of Theorem 2.2.
which completes the proof.
If we substitute , , and in the above results, then the results of generalized Stolarsky type means proved in  are recaptured.
This research was partially funded by Higher Education Commission, Pakistan. The research of the _rst author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant 117-1170889-0888.
- Pečarić JE, Proschan F, Tong YL: Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering. Volume 187. Academic Press, Boston, Mass, USA; 1992:xiv+467.Google Scholar
- Dragomir SS, Pearce CEM: Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monograph Collections. Victoria University; 2002.Google Scholar
- Vasić PM, Lacković IB: On an inequality for convex functions. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika 1974, (461–497):63–66.Google Scholar
- Lupaş A: A generalization of Hadamard inequalities for convex functions. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika 1976, (544–576):115–121.Google Scholar
- Vasić PM, Lacković IB: Some complements to the paper: "On an inequality for convex functions". Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika 1976, (544–576):59–62.Google Scholar
- Jakšetić J, Pečarić J, ur Rehman A: On Stolarsky and related means. Mathematical Inequalities and Applications 2010, 13: 899–909.MathSciNetGoogle Scholar
- Anwar M, Jakšetić J, Pečarić J, ur Rehman A: Exponential convexity, positive semi-definite matrices and fundamental inequalities. Journal of Mathematical Inequalities 2010, 4(2):171–189.MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.