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- Title
Optimal bounds for two Sándor-type means in terms of power means.
- Authors
Zhao, Tie-Hong; Qian, Wei-Mao; Song, Ying-Qing
- Abstract
In the article, we prove that the double inequalities $M_{\alpha }(a,b)< S_{QA}(a,b)< M_{\beta}(a,b)$ and $M_{\lambda }(a,b)< S_{AQ}(a,b)< M_{\mu}(a,b)$ hold for all $a, b>0$ with $a\neq b$ if and only if $\alpha\leq\log 2/[1+\log2-\sqrt{2}\log(1+\sqrt{2})]=1.5517\ldots$ , $\beta\geq5/3$, $\lambda\leq4\log2/[4+2\log2-\pi]=1.2351\ldots$ and $\mu\geq4/3$, where $S_{QA}(a,b)=A(a,b)e^{Q(a,b)/M(a,b)-1}$ and $S_{AQ}(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}$ are the Sándor-type means, $A(a,b)=(a+b)/2$, $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$, $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$, and $M(a,b)=(a-b)/[2\sinh ^{-1}((a-b)/(a+b))]$ are, respectively, the arithmetic, quadratic, second Seiffert, and Neuman-Sándor means.
- Subjects
ARITHMETICAL algebraic geometry; QUADRATIC equations; VON Neumann algebras; ARCCOSINE function; HYPERBOLIC functions
- Publication
Journal of Inequalities & Applications, 2016, Vol 2016, Issue 1, p1
- ISSN
1025-5834
- Publication type
Article
- DOI
10.1186/s13660-016-0989-0