Error function inequalities

We present various inequalities for the error function. One of our theorems states: Let α?≥?1. For all x,y?>?0 we have $$ delta_{alpha} < frac{ mbox{erf} left( x+ mbox{erf}(y)^{alpha}right) +mbox{erf}left( y+ mbox{erf}(x)^{alpha}right) } {mbox{erf}left( mbox{erf}(x)+mbox{erf}(y)right) } < Delta_{alpha} $$ with the best possible bounds $$ delta_{alpha}= left{ begin{array}{ll} 1+sqrt{pi}/2, & textrm{{if} $alpha=1$,} sqrt{pi}/2, & textrm{{if} $alpha>1$,} end{array}right. quad{mbox{and} ,,,,, Delta_{alpha}=1+frac{1}{mbox{erf}(1)}.} $$