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- W2923153527 abstract "In this article, we prove that the double inequalities $$begin{aligned} &alpha_{1} biggl[frac{7C(a,b)}{16}+frac{9H(a,b)}{16} biggr]+(1- alpha_{1}) biggl[frac{3A(a,b)}{4}+frac{G(a, b)}{4} biggr] &quad< E(a,b) &quad< beta_{1} biggl[frac{7C(a,b)}{16}+frac{9H(a,b)}{16} biggr]+(1- beta_{1}) biggl[frac{3A(a,b)}{4}+frac{G(a, b)}{4} biggr], &biggl[frac{7C(a,b)}{16}+frac{9H(a,b)}{16} biggr]^{alpha _{2}} biggl[ frac{3A(a,b)}{4}+frac{G(a, b)}{4} biggr]^{1-alpha_{2}} &quad< E(a,b) &quad< biggl[frac{7C(a,b)}{16}+frac{9H(a,b)}{16} biggr]^{beta _{2}} biggl[ frac{3A(a,b)}{4}+frac{G(a, b)}{4} biggr]^{1-beta_{2}} end{aligned}$$ hold for all $a, b>0$ with $aneq b$ if and only if $alpha_{1}leq 3/16=0.1875$ , $beta_{1}geq64/pi^{2}-6= 0.484555dots$ , $alpha_{2}leq3/16=0.1875$ and $beta_{2}geq(5log2-log3-2log pi)/(log7-log6)= 0.503817dots$ , where $E(a,b)= (frac{2}{pi}int^{pi/2}_{0}sqrt{acos^{2}theta +bsin^{2}theta},dtheta )^{2}$ , $H(a,b)=2ab/(a+b)$ , $G(a,b)=sqrt{ab}$ , $A(a,b)=(a+b)/2$ and $C(a,b)=(a^{2}+b^{2})/(a+b)$ are the quasi-arithmetic, harmonic, geometric, arithmetic and contra-harmonic means of a and b, respectively." @default.
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- W2923153527 date "2019-02-20" @default.
- W2923153527 modified "2023-10-16" @default.
- W2923153527 title "On approximating the quasi-arithmetic mean" @default.
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- W2923153527 doi "https://doi.org/10.1186/s13660-019-1991-0" @default.
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