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.


$\displaystyle \Delta _{n}^{}$(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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(Metadata: XML, RSS, BibTeX) MathSciNet: MR2199646 Z'blatt-MATH: 1095.39022


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