- Research Article
- Open Access
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.
- Convex Function
- Previous Theorem
- Positive Semidefinite
- Positive Weight
- Continuous Extension
It is easy to observe from the construction of the functions and that they do not depend essentially on , so we can write for short for , and so on.
where and are fixed integers.
By (1.6), and this implies that for . From (1.6), again, we have .
We need the following hypotheses.
(H2)Let and be positive -tuples such that .
(H3) Let be an interval, let , let be a positive -tuples such that , and let , be continuous and strictly monotone functions.
(H4) Let be an interval, let , and let be positive -tuples such that . Further, let be a convex function.
We deduce the monotonicity of these means from the following refinement of the discrete Jensen inequality in .
where the numbers are defined in (1.10) and (1.11). If is a concave function, then the inequalities in (1.16) are reversed.
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 .
When or , we get the required results by taking limit.
The monotonicity of these generalized means is obtained in the next corollary.
if either is convex and is decreasing or is concave and is increasing.
First, we can apply Theorem 1.1 to the function and the -tuples , then we can apply to the inequality coming from (1.16). This gives (1.23). A similar argument gives (1.24): and can be used.
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 .
If is a concave function then the inequalities (1.38) are reversed.
The proof comes from Corollary 1.2.
where either is convex and is decreasing or is concave and is increasing.
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 .
From now on, (H1) and (H4) are assumed if we consider , further, the hypothesis for any is also assumed if we consider . The numbers are generated by concrete examples, and (H4) is assumed.
We need the following subclass of convex functions (see ).
for all and all choices and .
We quote here useful propositions from .
Let be a function. Then, the following statements are equivalent
(i) is exponentially convex.
for every and every .
First, we introduce a special class of functions.
(H5) Let be a set of twice differentiable convex functions such that the function is exponentially convex for every fixed .
It is easy to see that the sets of functions and satisfy (H5).
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 .
Assume (H5), and suppose that the functions are continuous. The following statements hold for .
(b)The function is exponentially convex.
By taking in (2.1), we have
This means that the matrix is positive semidefinite, that is, (2.9) is valid.
(b)It is assumed that the function is continuous. By using Poposition 2.2 and (a), we get the exponential convexity of the function .
Since the functions are continuous , we have the following.
The function are exponentially convex. This remains valid if we replace by in (2.8).
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.
(a)Since the function , is invertible, then we get, from (3.3), that for
By choosing and , we can see that the expression between and in (3.4) defines a mean.
We can apply Theorem 3.2 to the functions , , , and , and the -tuples .
Suppose the conditions of Corollary 3.5 are satisfied.
(a)Since the function is invertible, then we get, from (3.5) and (3.6) that
(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.
This gives (3.10) if and . The other cases come from this by taking limit.
Now, we give the monotonicity of these new means.
If , set in (3.15) to obtain (3.14).
For , we get the required result by limit.
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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