A Cauchy-Type Functional Inequality for the Error Function

Abstract

Let erf be the error function. We prove that the functional inequality

$$(xy)^{\alpha}{\rm erf}(xy)^{\beta}\leq x^{\alpha}{\rm erf}(x)^{\beta} +y^{\alpha}{\rm erf}(y)^{\beta}\quad{(\alpha, \beta\in \mathbb{R})}$$

holds for all positive real numbers ${x}$ and ${y}$ if and only if either

$$\alpha=0 \quad {\rm and} \quad{0\leq \beta\leq \min_{1 < x\leq 1.4}\frac{\log 2}{\log \Delta(x)}=13.14249\ldots}$$

or

$$\alpha+\beta=0 \quad {\rm and}\quad{0 < \beta\leq \min_{0 < x < 1}\frac{\log 2}{\log \Delta(x)-\log{x}}=9.16124\ldots}.$$

Here, ${\Delta(x)={\rm erf}(x^2)/{\rm erf}(x)}$.