# Means and Their Inequalities by P. S. Bullen, D. S. Mitrinović, P. M. Vasić (auth.)

By P. S. Bullen, D. S. Mitrinović, P. M. Vasić (auth.)

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Extra resources for Means and Their Inequalities

Sample text

Remark ( 10) It is easily seen that this inequality is strict unless a' and aare identical. Theorem 17. Proof. 1: }"n-k (1-}") k(~), by Lemma 16, k"'O = 1. The general case follows using lemma 14. Remarks (11) Obviously (1 +~, 1 + ~ ,... / (1 + -2'-, ... n n n ~ n-l + -2'-, n-l 1) and so by Theorem 17 (1 + x )n-l < (1 +~)n/ n n=T this is a special case of inequality 3(8).

3. 'nleorem 22. If'l U + and convex then Proof. 3. l !. 4I

N wi f (a i i .. l The case of equality is easily considered. ~ " by the induction hypothesis • CHAPTERI 24 Remarks (7). 189; the above proof is repeated in see also McShane [1), Pop [1). This important result will be referred to a J. It is equivalent to convexity, as is seen from the above proof. (9) As we will see J includes most classical inequalities as special cases although this may not always be obvious. Wn Wn ~ Thus if W is a positive n-tple wi m i l wi follows fram the concavity of log.