## 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

### Keywords:

logarithmic convexity
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