A sharp lower bound for the complete elliptic integrals of the first kind
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A sharp lower bound for the complete elliptic integrals of the first kind Zhen-Hang Yang1,2 · Jing-Feng Tian3
· Ya-Ru Zhu3
Received: 19 April 2020 / Accepted: 8 October 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract Let K (r ) be the complete elliptic integrals of the first kind and arthr denote the inverse hyperbolic tangent function. We prove that the inequality 1/q 2 arthr q K (r ) > 1 − λ + λ π r holds for r ∈ (0, 1) with the best constants λ = 3/4 and q = 1/10. This improves some known results and gives a positive answer for a conjecture on the best upper bound for the Gaussian arithmetic–geometric mean in terms of logarithmic and arithmetic means. Keywords Arithmetic–geometric mean · Logarithmic mean · Complete elliptic integrals of the first kind · Inverse hyperbolic tangent function · NP type power series · Inequality Mathematics Subject Classification Primary 33E05 · 26E60; Secondary 40A99 · 41A21
Dedicated to the 60th anniversary of Zhejiang Electric Power Company Research Institute. This work was supported by the Fundamental Research Funds for the Central Universities under Grant 2015ZD29, Grant 13ZD19, and Grant MS117.
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Jing-Feng Tian [email protected] Zhen-Hang Yang [email protected] Ya-Ru Zhu [email protected]
1
Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education, North China Electric Power University, Yonghua Street 619, Baoding 071003, People’s Republic of China
2
Zhejiang Electric Power Company Research Institute, Hangzhou 310014, Zhejiang, People’s Republic of China
3
Department of Mathematics and Physics, North China Electric Power University , Yonghua Street 619, Baoding 071003, People’s Republic of China 0123456789().: V,-vol
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1 Introduction Let a, b > 0 with a = b. The Gaussian arithmetic–geometric mean (AGM) is defined by AG M (a, b) = lim an = lim bn , n→∞
n→∞
where a0 = a, b0 = b, and for n ∈ N, a n + bn (1.1) , bn+1 = G (an , bn ) = an bn . 2 An amazing connection between AG M (a, b) and complete elliptic integrals of the first kind K (r ) is given by Gauss’ formula π AG M 1, r = , (1.2) 2K(r ) an+1 = A (an , bn ) =
where
K (r ) =
π/2
0
1 1 − r 2 sin2 t
dt,
√ here and in what follows r = 1 − r 2 ∈ (0, 1) (see [1]). It is known that the complete elliptic integrals of the first kind K (r ) can be also represented by the Gaussian hypergeometric function
∞ 1 1 π (1/2)2n 2n K (r ) = F r , (1.3) , ; 1; r 2 = 2 2 2 (n!)2 n=0 where (a)0 = 1 for a = 0, (a)n = a(a + ∞1)(a + 2) · · · (a + n − 1) = (a + n)/ (a) is the shifted factorial function and (x) = 0 t x−1 e−t dt (x > 0) is the gamma function. The famous Landen identities [2, page 507] shows that √ 2 r 1−r 1 + r (1.4) K K r = (1 + r ) K (r ) and K = 1+r 1+r 2 for all r ∈ (0, 1). Moreover, an asymptotic formula for K (r ) as r → 1− is given by 4
K (r ) ∼ ln as r → 1− r
(1.5)
(see [2, page 299]). There is a number of bounds for the Gaussian arithmetic–geometric mean A
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