Functional inequalities for the error function. II (Q621358)

The error function, also known as probability integral, is defined for \(x\in\mathbb R\) by the integral representation \[ \text{erf}(x)= \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt. \] The aim of the paper under review is to prove the following functional inequalities for the error function \[ \text{erf}(1)<\frac{\text{erf}(x+\text{erf}(y))}{\text{erf}(y+\text{erf}(x))}<\frac{2}{\sqrt{\pi}} \qquad\text{and}\qquad 0<\frac{\text{erf}(x \operatorname{erf}(y))}{\text{erf}(y \operatorname{erf}(x))}\leq 1, \] where \(x\) and \(y\) are positive real numbers with \(x\leq y\), and obtained constant bounds are the best possible. To prove these results, the author obtains some lemmas which study monotonicity of some one variable functions, and also positivity of some two variable functions, all defined in terms of the error function. [Part I, cf. the author, ibid. 66, No. 1--2, 119--127 (2003; Zbl 1072.33001).]