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Optimal inequalities for the convex combination of error function. (English) Zbl 1314.33003
Summary: For $$\lambda \in (0,1)$$ and $$x,y > 0$$ we obtain the best possible constants $$p$$ and $$r$$, such that
$\operatorname{erf}(M_p(x,y;\lambda))\leq \lambda \operatorname{erf}(x) + (1 - \lambda) \operatorname{erf}(y)\leq\operatorname{erf}(M_r(x,y;\lambda))$ where $$\operatorname{erf}(x) = \frac 2 {\sqrt{\pi}}\int_0^x e^{- t^2} dt$$ and $$M_p(x,y;\lambda)=(\lambda x^p + (1-\lambda)y^p)^{1/p}$$ ($$p\neq 0$$), $$M_0(x,y;\lambda)=x^\lambda y^{1-\lambda}$$ are error function and weighted power mean, respectively. Furthermore, using these results, we generalized and complement an inequality due to Alzer.

##### 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
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