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Optimal lower and upper bounds for the geometric convex combination of the error function. (English) Zbl 1332.33003
Summary: For $$x\in\mathbb R$$, the error function $$\operatorname{erf}(x)$$ is defined as $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt.$ In this paper, we answer the question: what are the greatest value $$p$$ and the least value $$q$$, such that the double inequality $$\operatorname {erf}(M_{p}(x,y;\lambda))\leq G(\operatorname{erf}(x),\operatorname {erf}(y);\lambda)\leq\operatorname{erf}(M_{q}(x,y;\lambda))$$ holds for all $$x,y\geq1$$ (or $$0< x,y<1$$) and $$\lambda\in(0,1)$$? Here, $$M_{r}(x,y;\lambda)=(\lambda x^{r}+(1-\lambda)y^{r})^{1/r}$$ ($$r\neq0$$), $$M_{0}(x,y;\lambda)=x^{\lambda}y^{1-\lambda}$$ and $$G(x,y;\lambda )=x^{\lambda}y^{1-\lambda}$$ are the weighted power and the weighted geometric mean, respectively.

##### MSC:
 33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) 26D15 Inequalities for sums, series and integrals
##### Keywords:
error function; power mean; functional inequalities
DLMF; Equator
Full Text:
##### References:
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