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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
Software:
DLMF; Equator
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