Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means

Hong-Hu Chu; Tie-Hong Zhao; Yu-Ming Chu

doi:10.1515/ms-2017-0417

Published by De Gruyter September 27, 2020

Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means

Hong-Hu Chu , Tie-Hong Zhao and Yu-Ming Chu

From the journal Mathematica Slovaca

https://doi.org/10.1515/ms-2017-0417

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Abstract

In the article, we present the best possible parameters α₁, β₁, α₂, β₂ ∈ ℝ and α₃, β₃ ∈ [1/2, 1] such that the double inequalities

α1C(a,b)+(1−α1)A(a,b)<T3(a,b)<β1C(a,b)+(1−β1)A(a,b),α2C(a,b)+(1−α2)Q(a,b)<T3(a,b)<β2C(a,b)+(1−β2)Q(a,b),C(α3;a,b)<T3(a,b)<C(β3;a,b)

hold for a, b > 0 with a ≠ b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean, Q(a,b)=a2+b2/2 is the quadratic mean, C(a, b) = (a² + b²)/(a + b) is the contra-harmonic mean, C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] is the one-parameter contra-harmonic mean and T3(a,b)=(2π∫0π/2a3cos2⁡θ+b3sin2⁡θdθ)2/3 is the Toader mean of order 3.

Keywords: Toader mean; complete elliptic integral; arithmetic mean; quadratic mean; contraharmonic mean

MSC 2010: Primary 26E60; Secondary 33E05

This research was supported by the Natural Science Foundation of China (Grant Nos. 11971142, 61673169, 11871202, 11701176, 11626101, 11601485) and the Natural Science Foundation of Zhejiang Province (Grant No. LY19A010012).

(Communicated by Tomasz Natkaniec)

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Received: 2019-10-07

Accepted: 2020-01-14

Published Online: 2020-09-27

Published in Print: 2020-10-27

Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means

Abstract

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