A pair of optimal inequalities related to the error function

Elisabeth M. Werner, Mary Beth Ruskai

Abstract: The Error Function �egin{eqnarray} V(x) & equiv & sqrt{pi} e^{x^2} [1 - hbox{erf}(x)] \ & = & int_0^infty frac{ e^{-u} }{sqrt{x^2 + u}} du = 2 e^{x^2}int_x^infty e^{-t^2} dt onumber end{eqnarray} arises in many contexts, from probability to mathematical physics. We give estimates for the Error Function from above and below which are optimal within a certain class of functions.

Mathematics Subject Classification ID

Exponential and trigonometric functions (33B10)

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