A new upper bound in the second Kershaw's double inequality and its generalizations

Qi, Feng; Guo, Shuangjian; Chen, Shouxin

doi:10.1016/j.cam.2007.07.037

Cited by 18 publications

(13 citation statements)

References 16 publications

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“…it was claimed in [45,Remark 5] and [46,Remark 6] that the the identric mean I a,b (x) is a Bernstein function of x ∈ (− min{a, b}, ∞). See also [63, Section 1.5] and its preprint [61, Section 1.5].…”

Section: Origins and Motivationsmentioning

confidence: 99%

Integral Representations for Bivariate Complex Geometric Mean and Applications

Qi¹,

Lim²

2017

Preprint

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Abstract. In the paper, the authors survey integral representations (including the Lévy-Khintchine representations) and applications of some bivariate means (including the logarithmic mean, the identric mean, Stolarsky's mean, the harmonic mean, the (weighted) geometric means and their reciprocals, and the Toader-Qi mean) and the multivariate (weighted) geometric means and their reciprocals, derive integral representations of bivariate complex geometric mean and its reciprocal, and apply these newly-derived integral representations to establish integral representations of Heronian mean of power 2 and its reciprocal.

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Section: Origins and Motivationsmentioning

confidence: 99%

Integral Representations for Bivariate Complex Geometric Mean and Applications

Qi¹,

Lim²

2017

Preprint

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“…The topic of bounding the ratio of two gamma functions has a history of at least sixty years since [53]. For more information on its history, backgrounds, motivations and recent developments, please refer to, for example, [2,3,25,28,29,30,33,38,42,46,56,57], especially to the expository and survey preprint [31] in which plentiful references are collected. For knowledge of mean values, please refer to the celebrated book [13] or the paper [32].…”

Section: Logarithmically Completely Monotonic Function In X;mentioning

confidence: 99%

Complete monotonicity of some functions involving polygamma functions

Guo

2010

Journal of Computational and Applied Mathematics

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In the present paper, we establish necessary and sufficient conditions for the functions $x^\alpha\bigl\lvert\psi^{(i)}(x+\beta)\bigr\lvert$ and $\alpha\bigl\lvert\psi^{(i)}(x+\beta)\bigr\lvert-x\bigl\lvert\psi^{(i+1)}(x+\beta)\bigr\lvert$ respectively to be monotonic and completely monotonic on $(0,\infty)$, where $i\in\mathbb{N}$, $\alpha>0$ and $\beta\ge0$ are scalars, and $\psi^{(i)}(x)$ are polygamma functions.Comment: 14 page

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“…For more information, please refer to [5,7,9,10,11,12,14,19,20,21,22,23,30,32,34,35,36,38,39,40,41,49] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…In [41], the right hand side inequalities in (3) and (7) were refined as follows: For s, t ∈ R with s = t and x > − min{s, t}, inequalities (8) and…”

Section: Introductionmentioning

confidence: 99%