Refinements of Results about Weighted Mixed Symmetric Means and Related Cauchy Means
© László Horváth et al. 2011
Received: 26 November 2010
Accepted: 23 February 2011
Published: 14 March 2011
A recent refinement of the classical discrete Jensen inequality is given by Horváth and Pečarić. In this paper, the corresponding weighted mixed symmetric means and Cauchy-type means are defined. We investigate the exponential convexity of some functions, study mean value theorems, and prove the monotonicity of the introduced means.
1. Introduction and Preliminary Results
We need the following hypotheses.
We deduce the monotonicity of these means from the following refinement of the discrete Jensen inequality in .
Assume . To obtain (1.18), we can apply Theorem 1.1 to the function and the -tuples to get the analogue of (1.16) and to raise the power . Equation (1.19) can be proved in a similar way by using and and raising the power .
The monotonicity of these generalized means is obtained in the next corollary.
Throughout Examples 1.4-1.5, 1.9–1.12, which are based on examples in , the conditions (H2), in the mixed symmetric means, and (H3), in the quasiarithmetic means, will be assumed.
The following result is also given in .
The proof comes from Corollary 1.2.
The proof is a consequence of Corollary 1.3.
Equation (1.48) is a weighted mixed symmetric mean and (1.49) is a generalized mean, as given in . Therefore, Corollaries 1.7 and 1.8 are more general than the Corollaries 1.2 and 1.3 given in .
Equation (1.54) represents weighted mixed symmetric means, and (1.55) defines generalized means, as given in . Therefore, Corollaries 1.7 and 1.8 are more general than the Corollaries 1.9 and 1.10 given in .
2. Main Results
We need the following subclass of convex functions (see ).
We quote here useful propositions from .
First, we introduce a special class of functions.
In this paper we prove the exponential convexity of the functions , and we give mean value theorems for . We also define the respective means of Cauchy type and study their monotonicity. The results for are special cases of the results for , and the results for are special cases of results for . Especially, the results for are also given in .
3. Cauchy Means
In this section, first, we are interested in mean value theorems.
provided that the denominators are nonzero.
The idea of the proofs of Theorems 3.1 and 3.2 is the same as the proofs of Theorems 2.3 and 2.4 in .
Theorem 3.2 can be applied.
Suppose the conditions of Corollary 3.3 are satisfied.
Suppose the conditions of Corollary 3.5 are satisfied.
(b)As in Remark 3.4 (b), the expressions in (3.7) define means.
Now, we deduce the monotonicity of these means in the form of Dresher's inequality as follows.
Now, we give the monotonicity of these new means.
This research was partially funded by the Higher Education Commission, Pakistan.
The research of L. Horváth was supported by the Hungarian National Foundations for Scientific Research, Grant no. K73274, and that of J. Pecărić was supported by the Croatian Ministry of Science, Education, and Sports under the research Grant 117-1170889-0888.
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