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A note on differences of power means. (English) Zbl 1265.26084

Let \(\widetilde x_n=\{x_i\}^n_1\) and \(\widetilde p_n=\{p_i\}^n_1\) be two sequences of positive real numbers, \(\sum^n_1p_i=1\), and put \(d_m=d_m^{(n)}(\widetilde x_n,\widetilde p_n):=\sum_1^np_ix^m_i-(\sum_1^np_ix_i)^m\), \(m>1\). The main result of the paper is the following inequality: \(d_{m-1}d_{m+1}\geq c_m(d_m)^2\), \(m\geq3\). A nontrivial lower bound for \(d_m\) is given, too. The author also proves analogous inequalities with definite integrals, instead of finite sums.

MSC:

26D15 Inequalities for sums, series and integrals
26E60 Means
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