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Mean-value inequalities for the polygamma functions. (English) Zbl 0968.33003
The polygamma functions are derivatives of the logarithmic derivative $$\psi$$ of the gamma function. As one of the main results necessary and sufficient conditions are offered for the inequality $$|\psi^{(k)} [M^r (\mathbf{x},\mathbf{p})]|\leq M^s (|\psi^{(k)}(\mathbf{x)}|,\mathbf{p})$$ to hold for all $$\mathbf{x}=(x_1,\dots ,x_n)\in ]0,\infty[^n$$, $$\mathbf{p}=(p_1,\dots ,p_n)\in\{\mathbf{p}\in]0,\infty[^n\mid p_1 +\dots +p_n =1\}$$ $$k\geq 1, n\geq 2$$ fixed integers), where $$M^t(\mathbf{x},\mathbf{p})$$ is the $$t$$-th power mean of $$x_1,\dots ,x_n$$ with weights $$p_1,\dots ,p_n$$ (including the weighted geometric mean if $$t=0$$). If these conditions connecting $$r$$ and $$s$$ are satisfied then there is equality in the above inequality iff $$x_1=\dots =x_n.$$

##### MSC:
 33B15 Gamma, beta and polygamma functions 26D15 Inequalities for sums, series and integrals
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