Bull. Austral. Math. Soc. 72(3) pp.455--459, 2005.

A functional inequality for the
polygamma functions

Horst Alzer

Received: 25th July, 2005

I thank the referee for helpful comments.

Abstract

Let

$\displaystyle \Delta _{n}^{}$ (x) = $\displaystyle {\frac {{x^{n+1}}}{{n!}}}$ $\displaystyle \bigl \vert$ $\displaystyle \psi ^{{(n)}}_{}$ (x) $\displaystyle \bigr \vert$ (x > 0; n $\displaystyle \in$ N),

where $\psi$ denotes the logarithmic derivative of Euler's gamma function. We prove that the functional inequality

$\displaystyle \Delta _{n}^{}$ (x) + $\displaystyle \Delta _{n}^{}$ (y) < 1 + $\displaystyle \Delta _{n}^{}$ (z), x ^r +y ^r =z ^r ,

holds if and only if 0 < r $\leq$ 1. And, we show that the converse is valid if and only if r < 0 or r $\geq$ n + 1.

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MathSciNet: MR2199646

Z'blatt-MATH: 1095.39022

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ISSN 0004-9727

Bulletin of the Australian Mathematical Society

A functional inequality for the polygamma functions