A Note on Mixed-Mean Inequalities
© Peng Gao. 2010
Received: 25 February 2010
Accepted: 29 June 2010
Published: 14 July 2010
We give a simpler proof of a result of Holland concerning a mixed arithmetic-geometric mean inequality. We also prove a result of mixed-mean inequality involving the symmetric means.
Let be the generalized weighted power means: , where , , and , with . Here denotes the limit of as . Unless specified, we always assume that . When there is no risk of confusion, we will write for and we also define .
Now we focus our attention to the case of (1.2) for being weighted mean matrices given in (1.3). In this case, for fixed , we define , . Then we have the following mixed-mean inequalities of Nanjundiah  (see also ).
It is easy to see that the case of Theorem 1.1 follows from Theorem 1.2. As was pointed out by Kedlaya that the method used in  can be applied to establish both Popoviciu-type and Rado-type inequalities for mixed means for a general pair , the details were worked out in  and the following Rado-type inequalities were established in .
A different proof of Theorem 1.1 for the case was given in  and Bennett used essentially the same approach in [8, 9] to study (1.2) for the cases being lower triangular matrices, namely, if . Among other things, he showed  that inequalities (1.2) hold when are Hausdorff matrices.
In , Holland further improved the condition in Theorem 1.3 for the case by proving the following.
It is our goal in this paper to first give a simpler proof of the above result by modifying Holland's own approach. This is done in the next section and, in Section 3, we will prove a result of mixed-mean inequality involving the symmetric means.
2. A Proof of Theorem 1.4
Now it is easy to see that inequality (2.5) follows on adding inequalities (2.7) and (2.8), and this completes the proof of Theorem 1.4.
3. A Discussion on Symmetric Means
It is well known that, for fixed of dimension , is a nonincreasing function of for with (with weights , ). In view of the mixed-mean inequalities for the generalized weighted power means (Theorem 1.1), it is natural to ask whether similar results hold for the symmetric means. Of course one may have to adjust the notion of such mixed means in order for this to make sense for all . For example, when , , the notion of is not even defined. From now on, we will only focus on the extreme cases of the symmetric means; namely, or . In these cases it is then natural to define , and, on recasting , we see that it is also natural for us to define (note that this is not consistent with our definition of above).
We also need the following lemma of C. D. Tarnavas and D. D. Tarnavas .
We now apply Lemma 3.2 to obtain the following.
which is just what we want.
We now prove the following mixed-mean inequality involving the symmetric means.
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