## Rocky Mountain Journal of Mathematics

### Sub- and super-additive properties of the psi function

Horst Alzer

#### Abstract

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

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

Dates
First available in Project Euclid: 1 January 2015

https://projecteuclid.org/euclid.rmjm/1420071547

Digital Object Identifier
doi:10.1216/RMJ-2014-44-5-1399

Mathematical Reviews number (MathSciNet)
MR3295635

Zentralblatt MATH identifier
1308.33001

#### Citation

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. https://projecteuclid.org/euclid.rmjm/1420071547

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