On approximating the quasi-arithmetic mean

By Tie-hong Zhao, B. Zhou, Miao-Kun Wang, Y. Chu. Journal of Inequalities and Applications, 2019

76 citations recorded by the source corpus.

AbstractIn 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]<E(a,b)<β1[7C(a,b)16+9H(a,b)16]+(1−β1)[3A(a,b)4+G(a,b)4],[7C(a,b)16+9H(a,b)16]α2[3A(a,b)4+G(a,b)4]1−α2<E(a,b)<[7C(a,b)16+9H(a,b)16]β2[3A(a,b)4+G(a,b)4]1−β2 $$\begin{aligned} &\alpha_{1} \biggl[\frac{7C(a,b)}{16}+\frac{9H(a,b)}{16} \biggr]+(1- \alpha_{1}) \biggl[\frac{3A(a,b)}{4}+\frac{G(a, b)}{4} \biggr]\\ &\quad< E(a,b) \\ &\quad< \beta_{1} \biggl[\frac{7C(a,b)}{16}+\frac{9H(a,b)}{16} \biggr]+(1- \beta_{1}) \biggl[\frac{3A(a,b)}{4}+\frac{G(a, b)}{4} \biggr], \\ &\biggl[\frac{7C(a,b)}{16}+\frac{9H(a,b)}{16} \biggr]^{\alpha _{2}} \biggl[ \frac{3A(a,b)}{4}+\frac{G(a, b)}{4} \biggr]^{1-\alpha_{2}}\\ &\quad< E(a,b) \\ &\quad< \biggl[\frac{7C(a,b)}{16}+\frac{9H(a,b)}{16} \biggr]^{\beta _{2}} \biggl[ \frac{3A(a,b)}{4}+\frac{G(a, b)}{4} \biggr]^{1-\beta_{2}} \end{aligned}$$ hold for all a,b>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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b1d03b839b6f18f06bb32bee935a999a0c76e1f2

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In Applied Mathematics

  1. Tie-hong Zhao
  2. Hong-Hu Chu
  3. Yu‐ming Chu
  4. Y. Chu

In Geometry and Topology

  1. Tie-hong Zhao
  2. Hong-Hu Chu
  3. Yu‐ming Chu
  4. Y. Chu

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