Optimal inequalities for bounding Toader mean by arithmetic and quadratic means [PDF]
Journal of Inequalities and Applications, 2017 In this paper, we present the best possible parameters α ( r ) $\alpha(r)$ and β ( r ) $\beta(r)$ such that the double inequality [ α ( r ) A r ( a , b ) + ( 1 − α ( r ) ) Q r ( a , b ) ] 1 / r < T D [ A ( a , b ) , Q ( a , b ) ] < [ β ( r ) A r ( a , b )
Tie-Hong Zhao, Yu-Ming Chu, Wen Zhang
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Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean [PDF]
The Scientific World Journal, 2013 The authors find the greatest value λ and the least value μ, such that the double inequality C¯(λa+(1-λb),λb+(1-λ)a)
Wei-Dong Jiang
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Optimal bounds for Toader mean in terms of general means [PDF]
Journal of Inequalities and Applications, 2020 In this paper, we present the best possible parameters α ( r ) $\alpha (r)$ , β ( r ) $\beta (r)$ such that the double inequality [ α ( r ) M r ( a , b ) + ( 1 − α ( r ) ) N r ( a , b ) ] 1 / r < TD [ M ( a , b ) , N ( a , b ) ] < [ β ( r ) M r ( a , b )
Qian Zhang, Bing Xu, Maoan Han
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On Approximating the Toader Mean by Other Bivariate Means [PDF]
Journal of Function Spaces, 2019 In the article, we provide several sharp bounds for the Toader mean by use of certain combinations of the arithmetic, quadratic, contraharmonic, and Gaussian arithmetic-geometric means.
Jun-Li Wang +3 more
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Some Inequalities for Bounding Toader Mean [PDF]
Journal of Function Spaces and Applications, 2013 By finding linear relations among differences between two special means, the authors establish some inequalities for bounding Toader mean in terms of the arithmetic, harmonic, centroidal, and contraharmonic means.
Wen-Hui Li, Miao-Miao Zheng
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On approximating the modified Bessel function of the first kind and Toader-Qi mean [PDF]
Journal of Inequalities and Applications, 2016 In the article, we present several sharp bounds for the modified Bessel function of the first kind I 0 ( t ) = ∑ n = 0 ∞ t 2 n 2 2 n ( n ! ) 2 $I_{0}(t)=\sum_{n=0}^{\infty}\frac{t^{2n}}{2^{2n}(n!)^{2}}$ and the Toader-Qi mean T Q ( a , b ) = 2 π ∫ 0 π ...
Zhen-Hang Yang, Yu-Ming Chu
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Optimal convex combination bounds of geometric and Neuman means for Toader-type mean [PDF]
Journal of Inequalities and Applications, 2017 In this paper, we prove that the double inequalities α N Q A ( a , b ) + ( 1 − α ) G ( a , b ) < T D [ A ( a , b ) , G ( a , b ) ] < β N Q A ( a , b ) + ( 1 − β ) G ( a , b ) , λ N A Q ( a , b ) + ( 1 − λ ) G ( a , b ) < T D [ A ( a , b ) , G ( a , b ) ]
Yue-Ying Yang, Wei-Mao Qian
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New Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean [PDF]
Mathematics, 2021 Let Ipx be the modified Bessel function of the first kind of order p. The upper and lower bounds in the form of simple rational functions about cosht and (sinht)/t for the function I0x are obtained.
Ling Zhu
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Optimal bounds for arithmetic-geometric and Toader means in terms of generalized logarithmic mean [PDF]
Journal of Inequalities and Applications, 2017 In this paper, we find the greatest values α 1 , α 2 $\alpha_{1},\alpha_{2}$ and the smallest values β 1 , β 2 $\beta_{1},\beta_{2}$ such that the double inequalities L α 1 ( a , b ) < AG ( a , b ) < L β 1 ( a , b ) $L_{\alpha_{1}}(a,b)0$ with a ≠ b $a ...
Qing Ding, Tiehong Zhao
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Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean
Journal of Inequalities and Applications, 2016 In the article, we prove that the function r ↦ E ( r ) / S 9 / 2 − p , p ( 1 , r ′ ) $r\mapsto \mathcal{E}(r)/S_{9/2-p, p}(1, r')$ is strictly increasing on ( 0 , 1 ) $(0, 1)$ for p ≤ 7 / 4 $p\leq7/4$ and strictly decreasing on ( 0 , 1 ) $(0, 1)$ for p ∈
Zhen-Hang Yang, Yu-Ming Chu, Wen Zhang
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