Inequalities for certain means in two arguments

Yang, Zhen-Hang; Chu, Yu-Ming
September 2015
Journal of Inequalities & Applications;9/26/2015, Vol. 2015 Issue 1, p1
Academic Journal
In this paper, we present the sharp bounds of the ratios $U(a,b)/L_{4}(a,b)$, $P_{2}(a,b)/U(a,b)$, $NS(a,b)/P_{2}(a,b)$ and $B(a,b)/NS(a,b)$ for all $a, b>0$ with $a\neq b$, where $L_{4}(a,b)=[(b^{4}-a^{4})/(4(\log b-\log a))]^{1/4}$, $U(a,b)=(b-a)/[\sqrt{2}\arctan((b-a)/\sqrt{2ab})]$, $P_{2}(a,b)=[(b^{2}-a^{2})/(2\arcsin ((b^{2}-a^{2})/(b^{2}+a^{2})))]^{1/2}$, $NS(a,b)=(b-a)/[2\sinh ^{-1}((b-a)/(b+a))]$, $B(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}$, $A(a,b)=(a+b)/2$, $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$, and $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$.


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