Inequalities for certain means in two arguments
Inequalities for certain means in two arguments
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 $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$ .