Mean-value inequalities for the polygamma functions

Summary.

Let $k \geq 1$ be an integer and let $\psi^{(k)}$ be the k-th derivative of the psi function, $\psi = \Gamma'/ \Gamma$. Further, let¶$M_n^{[t]}(x_{\nu};p_{\nu}) = \left(\sum^n_{\nu = 1} p_{\nu}x_{\nu}^t\right)^{1/t} \quad{(t\neq{0})}, \quad{M_n^{[0]}(x_{\nu};p_{\nu}) = \prod^n_{\nu = 1} x_{\nu}^{p_{\nu}}}$¶be the power mean of order t of $x_1,\ldots,x_n$ with positive weights $p_1,\ldots,p_n$, and $\sum_{{\nu}=1}^{n} p_{\nu} = 1$. We determine all real numbers $\alpha, r,$, and s such that the inequalities¶¶$\left({M_{n}^{[1]}(x_{\nu};p_{\nu})}\over{M_{n}^{[0]}(x_{\nu};p_{\nu})} \right)^{\alpha} \leq {M_{n}^{[0]}(|{\psi^{(k)}(x_{\nu})}|;p_{\nu})\over|\psi^{(k)}(M_{n}^{[1]}(x_{\nu};p_{\nu}))|}$¶and¶$|\psi^{(k)}(M_n^{[r]}(x_{\nu};p_{\nu}))|\leq{M_n^{[s]} (|\psi^{(k)}(x_{\nu})|;p_{\nu})}$¶hold for all $x_{\nu}>0$ and $p_{\nu}>0$ $({\nu}=1,\ldots,n)$ with $\sum _{{\nu}=1}^{n} p_{\nu} = 1$. Our results sharpen and generalize known inequalities.