Rocky Mountain Journal of Mathematics

Sub- and super-additive properties of the psi function

Horst Alzer

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We prove the following sub- and super-additive properties of the psi function. \begin{enumerate} \item[(i)] The inequality \[ \psi\bigl( (x+y)^{\alpha} \bigr) \leq \psi(x^{\alpha})+\psi(y^{\alpha}) \quad{(\alpha \in\mathbf{R})} \] holds for all $x,y>0$ if and only if $\alpha\leq \alpha_0=-1.0266\ldots$\,. Here, $\alpha_0$ is given by \[ 2^{\alpha_0}=\inf_{t>0} \frac{\psi^{-1}\bigl( 2\psi(t) \bigr)}{t}=0.4908\ldots\, . \] \item[(ii)] The inequality \[ \psi(x^{\beta})+\psi(y^{\beta}) \leq \psi\bigl( (x+y)^{\beta} \bigr) \quad(\beta \in\mathbf{R}) \] is valid for all $x,y>0$ if and only if $\beta=0$. \end{enumerate}

Article information

Rocky Mountain J. Math., Volume 44, Number 5 (2014), 1399-1414.

First available in Project Euclid: 1 January 2015

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Zentralblatt MATH identifier

Primary: 33B15: Gamma, beta and polygamma functions 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx]

Psi function sub-additive super-additive inequalities convex concave


Alzer, Horst. Sub- and super-additive properties of the psi function. Rocky Mountain J. Math. 44 (2014), no. 5, 1399--1414. doi:10.1216/RMJ-2014-44-5-1399.

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