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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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