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Yuming Chu - One of the best experts on this subject based on the ideXlab platform.
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approximations for the Complete Elliptic Integral of the second hbox kind kind
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, 2021Co-Authors: Weimao Qian, Miao-kun Wang, Yuming ChuAbstract:In this paper, we present the best possible parameters $$\alpha _{1}$$ , $$\alpha _{2}$$ , $$\alpha _{3}$$ , $$\alpha _{4}$$ , $$\beta _{1}$$ , $$\beta _{2}$$ , $$\beta _{3}$$ , $$\beta _{4} \in {\mathbb {R}}$$ such that $$\begin{aligned} \frac{\alpha _{1}}{H(x, y)}+\frac{1-\alpha _{1}}{L(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{1}}{H(x, y)}+\frac{1-\beta _{1}}{L(x, y)},\\ \frac{\alpha _{2}}{H(x, y)}+\frac{1-\alpha _{2}}{P(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{2}}{H(x, y)}+\frac{1-\beta _{2}}{P(x, y)},\\ \frac{\alpha _{3}}{H(x, y)}+\frac{1-\alpha _{3}}{N S(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{3}}{H(x, y)}+\frac{1-\beta _{3}}{N S(x, y)},\\ \frac{\alpha _{4}}{H(x, y)}+\frac{1-\alpha _{4}}{T(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{4}}{H(x, y)}+\frac{1-\beta _{4}}{T(x, y)} \end{aligned}$$ hold for all $$x, y>0$$ with $$x \ne y$$ , where H(x, y), G(x, y), L(x, y), A(x, y), NS(x, y), P(x, y) and T(x, y) are respectively the harmonic, geometric, logarithmic, arithmetic, Neuman-Sandor, and first and second Seiffert means of two distinct positive numbers x and y, and $$\begin{aligned} V(x,y)=\pi G^{2}(x, y) /\left[ 2\int _{0}^{\pi /2}\sqrt{A^{2}(x,y) \cos ^{2}\varphi +G^{2}(x,y)\sin ^{2}\varphi }d\varphi \right] \end{aligned}$$ is a new Seiffert-like mean. As applications, some new inequalities for the Complete Elliptic Integral of the second kind are given.
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a sharp double inequality involving generalized Complete Elliptic Integral of the first kind
Math 2020 Vol. 5 Pages 4512-4528, 2020Co-Authors: Tiehong Zhao, Miao-kun Wang, Yuming ChuAbstract:In the article, we establish a sharp double inequality involving the ratio of generalized Complete Elliptic Integrals of the first kind, which is the improvement and generalization of some previously known results.
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approximation for the Complete Elliptic Integral of the first kind
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, 2020Co-Authors: Weimao Qian, Yuming ChuAbstract:In the article, we present several sharp upper and lower bounds for the Complete Elliptic Integral of the first kind in terms of inverse trigonometric and inverse hyperbolic functions. As consequences, some sharp bounds for the Gaussian arithmetic-geometric mean in terms of other bivariate means are also given.
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high accuracy asymptotic bounds for the Complete Elliptic Integral of the second kind
Applied Mathematics and Computation, 2019Co-Authors: Yuming Chu, Zhenhang Yang, Wen ZhangAbstract:Abstract In the article, we prove that the double inequality π 2 J ( r ′ ) − 51 π − 160 160 r 16 E ( r ) π 2 J ( r ′ ) − 5 π 3 × 2 31 r 16 holds for all r ∈ (0, 1), where E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 ( t ) d t is the Complete Elliptic Integral of the second kind, r ′ = ( 1 − r 2 ) 1 / 2 and J ( r ) = 51 r 2 + 20 r r + 50 r + 20 r + 51 16 ( 5 r + 2 r + 5 ) .
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monotonicity rule for the quotient of two functions and its application
Journal of Inequalities and Applications, 2017Co-Authors: Zhenhang Yang, Weimao Qian, Yuming Chu, Wen ZhangAbstract:In the article, we provide a monotonicity rule for the function $[P(x)+A(x)]/[P(x)+B(x)]$ , where $P(x)$ is a positive differentiable and decreasing function defined on $(-R, R)$ ( $R>0$ ), and $A(x)=\sum ^{\infty}_{n=n_{0}}a_{n}x^{n}$ and $B(x)=\sum^{\infty }_{n=n_{0}}b_{n}x^{n}$ are two real power series converging on $(-R, R)$ such that the sequence $\{a_{n}/b_{n}\}_{n=n_{0}}^{\infty}$ is increasing (decreasing) with $a_{n_{0}}/b_{n_{0}}\geq(\leq)\ 1$ and $b_{n}>0$ for all $n\geq n_{0}$ . As applications, we present new bounds for the Complete Elliptic Integral $\mathcal{E}(r)=\int_{0}^{\pi /2}\sqrt{1-r^{2}\sin^{2}t}\,dt$ ( $0< r<1$ ) of the second kind.
Miao-kun Wang - One of the best experts on this subject based on the ideXlab platform.
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approximations for the Complete Elliptic Integral of the second hbox kind kind
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, 2021Co-Authors: Weimao Qian, Miao-kun Wang, Yuming ChuAbstract:In this paper, we present the best possible parameters $$\alpha _{1}$$ , $$\alpha _{2}$$ , $$\alpha _{3}$$ , $$\alpha _{4}$$ , $$\beta _{1}$$ , $$\beta _{2}$$ , $$\beta _{3}$$ , $$\beta _{4} \in {\mathbb {R}}$$ such that $$\begin{aligned} \frac{\alpha _{1}}{H(x, y)}+\frac{1-\alpha _{1}}{L(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{1}}{H(x, y)}+\frac{1-\beta _{1}}{L(x, y)},\\ \frac{\alpha _{2}}{H(x, y)}+\frac{1-\alpha _{2}}{P(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{2}}{H(x, y)}+\frac{1-\beta _{2}}{P(x, y)},\\ \frac{\alpha _{3}}{H(x, y)}+\frac{1-\alpha _{3}}{N S(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{3}}{H(x, y)}+\frac{1-\beta _{3}}{N S(x, y)},\\ \frac{\alpha _{4}}{H(x, y)}+\frac{1-\alpha _{4}}{T(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{4}}{H(x, y)}+\frac{1-\beta _{4}}{T(x, y)} \end{aligned}$$ hold for all $$x, y>0$$ with $$x \ne y$$ , where H(x, y), G(x, y), L(x, y), A(x, y), NS(x, y), P(x, y) and T(x, y) are respectively the harmonic, geometric, logarithmic, arithmetic, Neuman-Sandor, and first and second Seiffert means of two distinct positive numbers x and y, and $$\begin{aligned} V(x,y)=\pi G^{2}(x, y) /\left[ 2\int _{0}^{\pi /2}\sqrt{A^{2}(x,y) \cos ^{2}\varphi +G^{2}(x,y)\sin ^{2}\varphi }d\varphi \right] \end{aligned}$$ is a new Seiffert-like mean. As applications, some new inequalities for the Complete Elliptic Integral of the second kind are given.
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a sharp double inequality involving generalized Complete Elliptic Integral of the first kind
Math 2020 Vol. 5 Pages 4512-4528, 2020Co-Authors: Tiehong Zhao, Miao-kun Wang, Yuming ChuAbstract:In the article, we establish a sharp double inequality involving the ratio of generalized Complete Elliptic Integrals of the first kind, which is the improvement and generalization of some previously known results.
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optimal combinations bounds of root square and arithmetic means for toader mean
Proceedings - Mathematical Sciences, 2012Co-Authors: Miao-kun WangAbstract:We find the greatest value α1 and α2, and the least values β1 and β2, such that the double inequalities α1S(a,b) + (1 − α1) A(a,b) 0 with a ≠ b. As applications, we get two new bounds for the Complete Elliptic Integral of the second kind in terms of elementary functions. Here, S(a,b) = [(a2 + b2)/2]1/2, A(a,b) = (a + b)/2, and \(T(a,b)=\frac{2}{\pi}\int\limits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}{\rm d}\theta\) denote the root-square, arithmetic, and Toader means of two positive numbers a and b, respectively.
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On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function
Abstract and Applied Analysis, 2011Co-Authors: Miao-kun WangAbstract:We prove that the double inequality (
Matti Vuorinen - One of the best experts on this subject based on the ideXlab platform.
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LANDEN INEQUALITIES FOR ZERO-BALANCED HYPERGEOMETRIC FUNCTIONS
2016Co-Authors: Slavko Simic, Matti VuorinenAbstract:Abstract. For zero-balanced Gaussian hypergeometric functions F (a, b; a+b;x), a, b> 0, we determine maximal regions of ab plane where well-known Landen identities for the Complete Elliptic Integral of the first kind turn on respective inequalities valid for each x ∈ (0, 1). Thereby an exhausting answer is given to the open problem from [AVV]
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landen inequalities for zero balanced hypergeometric functions
Abstract and Applied Analysis, 2012Co-Authors: Slavko Simic, Matti VuorinenAbstract:For zero-balanced Gaussian hypergeometric functions , , we determine maximal regions of plane where well-known Landen identities for the Complete Elliptic Integral of the first kind turn on respective inequalities valid for each . Thereby an exhausting answer is given to the open problem from the work by Anderson et al., 1990.
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landen inequalities for zero balanced hypergeometric functions
arXiv: Classical Analysis and ODEs, 2011Co-Authors: Slavko Simic, Matti VuorinenAbstract:For zero-balanced Gaussian hypergeometric functions $ F(a,b;a+b;x),$ $a,b>0,$ we determine maximal regions of $ab$ plane where well-known Landen identities for the Complete Elliptic Integral of the first kind turn on respective inequalities valid for each $x\in (0,1)$. Thereby an exhausting answer is given to an open problem.
Wen Zhang - One of the best experts on this subject based on the ideXlab platform.
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high accuracy asymptotic bounds for the Complete Elliptic Integral of the second kind
Applied Mathematics and Computation, 2019Co-Authors: Yuming Chu, Zhenhang Yang, Wen ZhangAbstract:Abstract In the article, we prove that the double inequality π 2 J ( r ′ ) − 51 π − 160 160 r 16 E ( r ) π 2 J ( r ′ ) − 5 π 3 × 2 31 r 16 holds for all r ∈ (0, 1), where E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 ( t ) d t is the Complete Elliptic Integral of the second kind, r ′ = ( 1 − r 2 ) 1 / 2 and J ( r ) = 51 r 2 + 20 r r + 50 r + 20 r + 51 16 ( 5 r + 2 r + 5 ) .
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on approximating the arithmetic geometric mean and Complete Elliptic Integral of the first kind
Journal of Mathematical Analysis and Applications, 2018Co-Authors: Zhenhang Yang, Weimao Qian, Wen ZhangAbstract:Abstract In the article, we prove that the double inequalities 1 + ( 6 p − 7 ) r ′ p + ( 5 p − 6 ) r ′ π tanh − 1 ( r ) 2 r K ( r ) 1 + ( 6 q − 7 ) r ′ q + ( 5 q − 6 ) r ′ π tanh − 1 ( r ) 2 r , q A ( 1 , r ) + ( 5 q − 6 ) G ( 1 , r ) A ( 1 , r ) + ( 6 q − 7 ) G ( 1 , r ) L ( 1 , r ) A G M ( 1 , r ) p A ( 1 , r ) + ( 5 p − 6 ) G ( 1 , r ) A ( 1 , r ) + ( 6 p − 7 ) G ( 1 , r ) L ( 1 , r ) hold for all r ∈ ( 0 , 1 ) if and only if p ≥ π / 2 = 1.570796 ⋯ and q ≤ 89 / 69 = 1.289855 ⋯ , where K ( r ) = ∫ 0 π / 2 ( 1 − r 2 sin 2 t ) − 1 / 2 d t is the Complete Elliptic Integral of the first kind, tanh − 1 ( r ) = log [ ( 1 + r ) / ( 1 − r ) ] / 2 is the inverse hyperbolic tangent function, r ′ = 1 − r 2 , and A ( 1 , r ) = ( 1 + r ) / 2 , G ( 1 , r ) = r , L ( 1 , r ) = ( r − 1 ) / log r and A G M ( 1 , r ) are the arithmetic, geometric, logarithmic and Gaussian arithmetic-geometric means of 1 and r , respectively.
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monotonicity rule for the quotient of two functions and its application
Journal of Inequalities and Applications, 2017Co-Authors: Zhenhang Yang, Weimao Qian, Yuming Chu, Wen ZhangAbstract:In the article, we provide a monotonicity rule for the function $[P(x)+A(x)]/[P(x)+B(x)]$ , where $P(x)$ is a positive differentiable and decreasing function defined on $(-R, R)$ ( $R>0$ ), and $A(x)=\sum ^{\infty}_{n=n_{0}}a_{n}x^{n}$ and $B(x)=\sum^{\infty }_{n=n_{0}}b_{n}x^{n}$ are two real power series converging on $(-R, R)$ such that the sequence $\{a_{n}/b_{n}\}_{n=n_{0}}^{\infty}$ is increasing (decreasing) with $a_{n_{0}}/b_{n_{0}}\geq(\leq)\ 1$ and $b_{n}>0$ for all $n\geq n_{0}$ . As applications, we present new bounds for the Complete Elliptic Integral $\mathcal{E}(r)=\int_{0}^{\pi /2}\sqrt{1-r^{2}\sin^{2}t}\,dt$ ( $0< r<1$ ) of the second kind.
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monotonicity rule for the quotient of two functions and its application
Journal of Inequalities and Applications, 2017Co-Authors: Zhenhang Yang, Weimao Qian, Wen ZhangAbstract:In the article, we provide a monotonicity rule for the function $[P(x)+A(x)]/[P(x)+B(x)]$ , where $P(x)$ is a positive differentiable and decreasing function defined on $(-R, R)$ ( $R>0$ ), and $A(x)=\sum ^{\infty}_{n=n_{0}}a_{n}x^{n}$ and $B(x)=\sum^{\infty }_{n=n_{0}}b_{n}x^{n}$ are two real power series converging on $(-R, R)$ such that the sequence $\{a_{n}/b_{n}\}_{n=n_{0}}^{\infty}$ is increasing (decreasing) with $a_{n_{0}}/b_{n_{0}}\geq(\leq)\ 1$ and $b_{n}>0$ for all $n\geq n_{0}$ . As applications, we present new bounds for the Complete Elliptic Integral $\mathcal{E}(r)=\int_{0}^{\pi /2}\sqrt{1-r^{2}\sin^{2}t}\,dt$ ( $0< r<1$ ) of the second kind.
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optimal inequalities for bounding toader mean by arithmetic and quadratic means
Journal of Inequalities and Applications, 2017Co-Authors: Tiehong Zhao, Wen ZhangAbstract:In this paper, we present the best possible parameters $\alpha(r)$ and $\beta(r)$ such that the double inequality $$\begin{aligned} \bigl[\alpha(r)A^{r}(a,b)+ \bigl(1-\alpha(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} < & TD \bigl[A(a,b), Q(a,b) \bigr] \\ < & \bigl[\beta(r)A^{r}(a,b)+ \bigl(1-\beta(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} \end{aligned}$$ holds for all $r\leq 1$ and $a, b>0$ with $a\neq b$ , and we provide new bounds for the Complete Elliptic Integral $\mathcal{E}(r)=\int_{0}^{\pi/2}(1-r^{2}\sin^{2}\theta)^{1/2}\,d\theta$ $(r\in (0, \sqrt{2}/2))$ of the second kind, where $TD(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}\,d\theta$ , $A(a,b)=(a+b)/2$ and $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$ are the Toader, arithmetic, and quadratic means of a and b, respectively.
Slavko Simic - One of the best experts on this subject based on the ideXlab platform.
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LANDEN INEQUALITIES FOR ZERO-BALANCED HYPERGEOMETRIC FUNCTIONS
2016Co-Authors: Slavko Simic, Matti VuorinenAbstract:Abstract. For zero-balanced Gaussian hypergeometric functions F (a, b; a+b;x), a, b> 0, we determine maximal regions of ab plane where well-known Landen identities for the Complete Elliptic Integral of the first kind turn on respective inequalities valid for each x ∈ (0, 1). Thereby an exhausting answer is given to the open problem from [AVV]
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landen inequalities for zero balanced hypergeometric functions
Abstract and Applied Analysis, 2012Co-Authors: Slavko Simic, Matti VuorinenAbstract:For zero-balanced Gaussian hypergeometric functions , , we determine maximal regions of plane where well-known Landen identities for the Complete Elliptic Integral of the first kind turn on respective inequalities valid for each . Thereby an exhausting answer is given to the open problem from the work by Anderson et al., 1990.
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landen inequalities for zero balanced hypergeometric functions
arXiv: Classical Analysis and ODEs, 2011Co-Authors: Slavko Simic, Matti VuorinenAbstract:For zero-balanced Gaussian hypergeometric functions $ F(a,b;a+b;x),$ $a,b>0,$ we determine maximal regions of $ab$ plane where well-known Landen identities for the Complete Elliptic Integral of the first kind turn on respective inequalities valid for each $x\in (0,1)$. Thereby an exhausting answer is given to an open problem.