Schur-Convexity of Averages of Convex Functions
© Vera Čuljak et al. 2011
Received: 12 November 2010
Accepted: 11 January 2011
Published: 18 January 2011
The object is to give an overview of the study of Schur-convexity of various means and functions and to contribute to the subject with some new results. First, Schur-convexity of the generalized integral and weighted integral quasiarithmetic mean is studied. Relation to some already published results is established, and some applications of the extended result are given. Furthermore, Schur-convexity of functions connected to the Hermite-Hadamard inequality is investigated. Finally, some results on convexity and Schur-convexity involving divided difference are considered.
The property of Schur-convexity and Schur-concavity has invoked the interest of many researchers and numerous papers have been dedicated to the investigation of it. The object of this paper is to present an overview of the results related to the study of Schur-convexity of various means and functions, in particular, those connected with the Hermite-Hadamard inequality. Moreover, we contribute to the subject with some new results.
First, let us recall the definition of Schur-convexity. It generalizes the definition of the convex and concave function via the notion of majorization.
Note that every function that is convex and symmetric is also Schur-convex.
One of the references which will be of particular interest in this paper is . The authors were inspired by some inequalities concerning gamma and digamma function and proved the following result for the integral arithmetic mean.
Few years later, Wulbert, in , proved that the integral arithmetic mean defined in (1.3) is convex on if is convex on . Zhang and Chu, in , rediscovered (without referring to and citing Wulbert's result) that the necessary and sufficient condition for the convexity of the integral arithmetic mean is for to be convex on . Note that the necessity is obvious. Namely, if is convex, then it is also Schur-convex since it is symmetric. Theorem A1 then implies the convexity of function .
Later, in , the Schur-convexity of the weighted integral arithmetic mean was proved.
In the same reference, the authors left an open problem: under which conditions does (1.5) hold?
The monotonicity of the function defined in (1.4) was studied in .
In the following sections, Schur-convexity of the generalized integral and weighted integral quasiarithmetic mean is studied. Relation to some already published results is established. Further, a new proof of sufficiency in Theorem A1, which is also a new proof of Wulbert's result from , that is, Zhang and Chu's result from , is presented. Some applications of this extended result are given. Furthermore, Schur-convexity of various functions connected to the Hermite-Hadamard inequality is investigated. Finally, some results on convexity and Schur-convexity involving divided difference are considered.
To complete the Introduction, we state three very interesting lemmas related to Schur-convexity. They are needed later for the proofs of our results. All three can be found in both [6, 7]. The first one gives a useful characterization of Schur-convexity.
The next result discovers the property of Schur-convexity of the generalized integral quasiarithmetic means.
the following hold:
is Schur-convex for and Schur-concave for . This was also obtained in  as a consequence of Theorem A1.
the following hold:
Combining (a), (b), (a'), and (b') completes the proof.
(i) is Schur-convex on if is increasing and is increasing and convex and is convex or if is increasing and is decreasing and concave and is convex or if is decreasing and is decreasing and concave and is concave or if is decreasing and is increasing and convex and is concave,
(ii) is Schur-concave on if is decreasing and is decreasing and convex and is convex or if is decreasing and is increasing and concave and is convex or if is increasing and is increasing and concave and is concave or if is increasing and is decreasing and convex and is concave.
2.1. Application of Theorem A1 for the Extended Mean Values
Many properties of extended mean values have been considered in . It was shown that are continuous on and symmetric with respect to both and , and and .
Shi et al. in , using Theorem A1 and Lemma A3 obtained the following condition for the Schur-convexity of the extended mean values with respect to .
Chu and Zhang in  established the necessary and sufficient conditions for the extended mean values to be Schur-convex (Schur-concave) with respect to , for fixed .
We remark that the above result does not cover the case , that is, the case of the identric mean of order . Monotonicity and Schur-concavity of the identric mean with respect to and for fixed was discussed in , using the hyperbolic composite function.
The following result is an extension of Wulbert's result from .
We recall the following definitions and remarks (see, e.g., ).
Now, we will give some applications of (3.1).
Analogous to the proof of Theorem 2.2 from .
Following the steps of the proofs of Theorems 2.4 and 2.5 given in , we can prove the following two mean value theorems.
provided that denominator on right-hand side is nonzero.
It follows the steps of the proof of Theorem 4.2 given in .
As a special case for , , we recapture the discrete version of the results obtained in.
4. Hermite-Hadamard Inequality
In , it was shown that is convex if and only if at least one of the inequalities in (4.1) is valid.
An interesting fact is that the original proof of Theorem A1 was given using the second Hermite-Hadamard inequality and the first one follows from the same theorem.
The same inequality was rediscovered later in  through an elementary analytic proof.
4.1. Application of Theorem A1 for a Function Connected with Hadamard Inequality
the following hold:
the following hold:
Shi, in , found a similar result as Theorem A1 for this function .
4.2. Schur-Convexity of Hermite-Hadamard Differences
In , the property of Schur-convexity of the difference between the middle part and the left-hand side of the Hermite-Hadamard inequality (4.1), and the difference between the right-hand side and the middle part of the same inequality, was investigated. The following theorems were proved.
First, we state a simple consequence of Theorems A1, A10, and A11.
which, after applying Lemma A1, is another proof of (ii)⇔(iii) in Corollary 4.1.
Identity (4.13) enables us to give a new proof of sufficiency in Theorem A11.
With the help of identity (4.16), we can present the following.
is Schur-convex (Schur-concave).
5. Convexity and Schur-Convexity of Divided Differences
In this final section, we turn our attention towards divided differences. Let us first recall the definition.
For more details on divided differences and -convexity, see .
In , Zwick proved the following theorem.
Note that the divided difference is a permutation symmetric function. Thus, the following theorem follows from Theorem A13 and a result on majorization inequalities. It was obtained in  by Pečari and Zwick.
Many more results involving divided differences were obtained, among others the multivariate analogues, all of which can be found in .
About a decade later, Merkle in  presented the following.
Then, the conditions (A)–(E) are equivalent and the conditions (A)–(E') are equivalent, where
Thus, it becomes clear that the statements (A)⇔(E) and (A')⇔(E') are in fact an alternative statement of Theorem A1. Furthermore, implications (A)⇒(E) and (A')⇒(E') are a special case of Theorem A14, while (A)⇒(D) and (A')⇒(D') are a special case of Theorem A13.
Moreover, note that (B) and (C), that is, (B') and (C'), are in fact the Hermite-Hadamard inequalities and we have already commented on their relation with Theorem A1—one side is used in the proof and the other is a consequence of the theorem.
Furthermore, the statements (A)⇔(D) and (A')⇔(D') are an alternative statement of Zhang and Chu's result from  and the necessity part recaptures Wulbert's result from  and the result from our Theorem 3.1.
5.1. Applications of Schur-Convexity of Divided Differences
are obtained as special cases of this new mean. In , necessary conditions under which Gini means (5.11) are Schur-convex and Schur-concave were given. In the short note , Witkowski completed this result with the proof of sufficiency of those conditions.
Using this form and Theorem A15, he proved the following.
The following conditions are equivalent:
Witkowski was able to apply all the results obtained for the two-parameter means, in particular Theorem A16, for this new family of means. The one of special interest to us is the following.
Note that the same four-parameter family of means was the object of interest to Yang in . He gave conditions under which are increasing (decreasing) and logarithmically convex (logarithmically concave). Necessary and sufficient conditions for to be increasing (decreasing) were, however, given in .
The research of the authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grants nos. 117-1170889-0888 (for V. Čuljak and J. Pečarić) and 058-1170889-1050 (for I. Franjić).
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