A power mean inequality involving the complete elliptic integrals

Wang, Gendi; Zhang, Xiaohui; Chu, Yu‐Ming

doi:10.1216/rmj-2014-44-5-1661

Cited by 39 publications

(8 citation statements)

References 16 publications

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“…Recently, G.-D. Wang, X.-H. Zhang, and Y.-M. Chu [184,185] have extended these inequalities as follows: Some similar results for the generalized Grötzsch function, generalized modular function, and other special functions related to quasiconformal analysis can be found in [152,182,186,187,188].…”

Section: Means and Quasiconformal Analysismentioning

confidence: 93%

Topics in Special Functions III

Anderson

Вуоринен

Zhang

2014

Analytic Number Theory, Approximation Theory, and Special Functions

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The study of quasiconformal maps led the first two authors in their joint work with M. K. Vamanamurthy to formulate open problems or questions involving special functions [14,16]. During the past two decades, many authors have contributed to the solution of these problems. However, most of the problems posed in [14] are still open.The present paper is the third in a series of surveys by the first two authors, the previous papers [19,21] being written jointly with the late M. K. Vamanamurthy. The aim of this series of surveys is to review the results motivated by the problems in [14,16] and related developments during the past two decades. In the first of these we studied classical special functions, and in the next we focused on special functions occurring in the distortion theory of quasiconformal maps. Regretfully, Vamanamurthy passed away in 2009, and the remaining authors dedicate the present work as a tribute to his memory. For an update to the bibliographies of [19] and [21] the reader is referred to [23].In 1993 the following monotone rule was derived [17, Lemma 2.2]. Though simple to state and easy to prove by means of the Cauchy Mean Value Theorem, this l'Hôpital Monotone Rule (LMR) has had wide application to special functions by many authors. Vamanamurthy was especially skillful in the application of this rule. We here quote the rule as it was restated in [20, Theorem 2].

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Section: Means and Quasiconformal Analysismentioning

confidence: 93%

Topics in Special Functions III

Anderson

Вуоринен

Zhang

2014

Analytic Number Theory, Approximation Theory, and Special Functions

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“…A bivariate function Ω: (0, ∞) × (0, ∞) ⟶ (0, ∞) is said to be a mean if min a, b { } ≤ Ω(a, b) ≤ max a, b { } for all a, b ∈ (0, ∞). Recently, the bivariate means have been the subject of intensive research [63][64][65][66][67][68][69][70][71][72][73][74][75]; in particular, many remarkable inequalities and properties for the bivariate means and their related special functions can be found in the literature [76][77][78][79][80][81][82][83][84][85].…”

Section: Applications To Special Meansmentioning

confidence: 99%

Inequalities Involving Conformable Approach for Exponentially Convex Functions and Their Applications

Rashid

Liu

et al. 2020

Journal of Function Spaces

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“…It is not difficult to verify that the νth Hölder mean H ν (σ , τ ) is strictly increasing with respect to ν ∈ (-∞, ∞) for all distinct positive real numbers σ and τ , and are the classical harmonic, geometric, arithmetic, and quadratic means of σ and τ , respectively. The bivariate means have in the past decades been the subject of intense research activity [4][5][6][7][8][9][10][11][12][13] because many important special functions can be expressed by the bivariate means [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] and they have wide applications in mathematics, statistics, physics, economics , and many other natural and human social sciences .…”

Section: Introductionmentioning

confidence: 99%