- Research Article
- Open Access
Some Comparison Inequalities for Generalized Muirhead and Identric Means
© Miao-Kun Wang et al. 2010
- Received: 21 December 2009
- Accepted: 23 January 2010
- Published: 28 January 2010
- Real Number
- Simple Computation
- Intensive Research
- Classified Region
- Sharp Inequality
The generalized Muirhead mean was introduced by Trif , the monotonicity of with respect to or was discussed, and a comparison theorem and a Minkowski-type inequality involving the generalized Muirhead mean were discussed.
It is easy to see that the generalized Muirhead mean is continuous on the domain and differentiable with respect to for fixed with . It is symmetric in and and in and . Many means are special cases of the generalized Muirhead mean, for example,
The well-known Muirhead inequality  implies that if are fixed, then is Schur convex on the domain and Schur concave on the domain . Chu and Xia  discussed the Schur convexity and Schur concavity of with respect to for fixed with .
The main properties of the power mean are given in . In particular, is continuous and increasing with respect to for fixed . Let ,
The following sharp inequality is due to Carlson :
Pittenger  proved that
Alzer and Qiu  proved the inequalities
In , Chu and Xia proved that
In order to prove Theorem 1.1 we need Lemma 2.1 that follows.
then the following statements hold.
Proof of Theorem 1.1.
Without loss of generality, we assume that . From the symmetry we clearly see that Theorem 1.1 is true if we prove that is positive, negative, and neither positive nor negative with respect to for , and .
We divide the proof into three cases.
Equations (3.3)–(3.8) imply that
From (2.2) and (3.15) we clearly see that
On the other hand, from (3.3) we clearly see that
On the other hand, from (3.3) we clearly see that
This research is partly supported by N. S. Foundation of China under grant no. 60850005 and the N. S. Foundation of Zhejiang Province under grants no. D7080080 and no. Y607128.
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