Differences of Weighted Mixed Symmetric Means and Related Results
© Khuram Ali Khan et al. 2010
Received: 22 June 2010
Accepted: 13 October 2010
Published: 18 October 2010
Some improvements of classical Jensen's inequality are used to define the weighted mixed symmetric means. Exponential convexity and mean value theorems are proved for the differences of these improved inequalities. Related Cauchy means are also defined, and their monotonicity is established as an application.
1. Introduction and Preliminary Results
We obtain the monotonicity of these means as a consequence of the following improvement of Jensen's inequality .
In fact Corollaries 1.2 and 1.3 are weighted versions of results in .
The following result is valid [5, page 8].
The following result is given in [4, page 90].
The following result is given at [4, page 97].
The exponentially convex functions are defined in  as follows.
We also quote here a useful propositions from .
In this paper we prove the exponential convexity of (1.43) for convex functions defined in (1.46) and (1.47) and mean value theorems for the differences given in (1.43). We also define the corresponding means of Cauchy type and establish their monotonicity.
2. Main Result
The following theorems are the generalizations of results given in .
It was proved in  that is continuous for . By using Proposition 1.19, we get exponential convexity of the function .
provided that the denominators are non zero.
3. Cauchy Mean
The following lemma gives an equivalent definition of the convex function [4, page 2].
Now, we deduce the monotonicity of means given in (3.2) in the form of Dresher's inequality, as follows.
This together with (2.1) follows (3.5).
Now, we give the monotonicity of new means given in (3.10), as follows:
This research was partially funded by Higher Education Commission, Pakistan. The research of the second author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant no. 117-1170889-0888.
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