Monotonicity theorems and inequalities for the complete elliptic integrals

Alzer, Horst; Qiu, Song-Liang

doi:10.1016/j.cam.2004.02.009

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Introduction4

Corollary 2131

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2007

2022

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Cited by 159 publications

(84 citation statements)

References 26 publications

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“…In particular, µ a (r) satisfies the reverse Turán type inequality, that is for all a 1 , a 2 , r ∈ (0, 1) we have µ a 1 +a 2 2 (r) 2 ≤ µ a 1 (r)µ a 2 (r).…”

Section: Corollary 213mentioning

confidence: 99%

Turán type inequalities for generalized complete elliptic integrals

Baricz

2007

Math. Z.

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In this paper our aim is to establish some Turán type inequalities for Gaussian hypergeometric functions and for generalized complete elliptic integrals. These results complete the earlier result of P. Turán proved for Legendre polynomials. Moreover we show that there is a close connection between a Turán type inequality and a sharp lower bound for the generalized complete elliptic integral of the first kind. At the end of this paper we prove a recent conjecture of T. Sugawa and M. Vuorinen related to estimates of the hyperbolic distance of the twice punctured plane.

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“…In particular, µ a (r) satisfies the reverse Turán type inequality, that is for all a 1 , a 2 , r ∈ (0, 1) we have µ a 1 +a 2 2 (r) 2 ≤ µ a 1 (r)µ a 2 (r).…”

Section: Corollary 213mentioning

confidence: 99%

Turán type inequalities for generalized complete elliptic integrals

Baricz

2007

Math. Z.

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“…The complete elliptic integrals play a very important role in the study of conformal invariants [7], quasiconformal mappings [5,6,7,12] and Ramanujan's modular equations [4]. Numerous sharp inequalities and elementary approximations for the complete elliptic integrals have been proved in [2,3,7,8,13,14]. The special function m(r) is defined as…”

Section: Introductionmentioning

confidence: 99%

A power mean inequality involving the complete elliptic integrals

Wang¹,

Zhang²,

Chu³

2014

Rocky Mountain J. Math.

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“…This conjecture was proved by Qiu and Shen [26], and Barnard et al [3], respectively. Alzer and Qiu [1] presented a best possible upper power mean bound of the Toader mean as follows…”

Section: Introductionmentioning

confidence: 99%