Functional inequalities for the error function, II

Summary.

Let

$${\rm erf}(x) = \frac{2} {\sqrt{\pi}} \int_0^x e^{-t^{2}} dt$$

be the error function. We prove that the following inequalities are valid for all positive real numbers x, y with $x \leq y$:

$${\rm erf}(1) < \frac {{\rm erf} (x + {\rm erf}(y))} {{\rm erf}(y + {\rm erf}(x))} < \frac{2} {\sqrt{\pi}} \quad {\rm and} \quad 0 < \frac{{\rm erf}(x\, {\rm erf}(y))} {{\rm erf}(y\, {\rm erf}(x))} \leq 1$$

. The given constant bounds are the best possible.