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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New Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean
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 Toader mean in terms of general means
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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Optimal convex combination bounds of geometric and Neuman means for Toader-type mean
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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On approximating the modified Bessel function of the first kind and Toader-Qi mean
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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New Sharp Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean
Mathematics, 2020 Let I v x be he modified Bessel function of the first kind of order v. We prove the double inequality sinh t t cosh 1 / q q t
Zhen-Hang Yang +2 more
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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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Optimal bounds for arithmetic-geometric and Toader means in terms of generalized logarithmic mean
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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Sharp Generalized Seiffert Mean Bounds for Toader Mean
Abstract and Applied Analysis, 2011 For p∈[0,1], the generalized Seiffert mean of two positive numbers a and b is defined by Sp(a,b)=p(a-b)/arctan[2p(a-b)/(a+b ...
Yu-Ming Chu +3 more
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