On approximating the quasi-arithmetic mean

Abstract In this article, we prove that the double inequalities α1[7C(a,b)16+9H(a,b)16]+(1−α1)[3A(a,b)4+G(a,b)4]0 $a, b>0$ with a≠b $a\neq b$ if and only if α1≤3/16=0.1875 $\alpha_{1}\leq 3/16=0.1875$, β1≥64/π2−6=0.484555… $\beta_{1}\geq64/\pi^{2}-6= 0.484555\dots$, α2≤3/16=0.1875 $\alpha_{2}\leq3/16=0.1875$ and β2≥(5log2−log3−2logπ)/(log7−log6)=0.503817… $\beta_{2}\geq(5\log2-\log3-2\log \pi)/(\log7-\log6)= 0.503817\dots$, where E(a,b)=(2π∫0π/2acos2θ+bsin2θdθ)2 $E(a,b)= (\frac{2}{\pi}\int^{\pi/2}_{0}\sqrt{a\cos^{2}\theta +b\sin^{2}\theta}\,d\theta )^{2}$, H(a,b)=2ab/(a+b) $H(a,b)=2ab/(a+b)$, G(a,b)=ab $G(a,b)=\sqrt{ab}$, A(a,b)=(a+b)/2 $A(a,b)=(a+b)/2$ and C(a,b)=(a2+b2)/(a+b) $C(a,b)=(a^{2}+b^{2})/(a+b)$ are the quasi-arithmetic, harmonic, geometric, arithmetic and contra-harmonic means of a and b, respectively