Journal ArticleDOI
Integral representation of some functions related to the Gamma function
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In this paper, it was shown that the functions Θ(Θ(x) = [Gamma (x + 1)]^{1/x} (1 + 1/x)^x /x) and Θ (Θ (x, Θ) = Θ((Θ + 1)/x)) are Stieltjes transforms.Citations
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A Family of Horn-Bernstein Functions
TL;DR: In this article, a family of recently investigated Bernstein functions is revisited and those functions for which the derivatives are logarithmically completely monotonic are identified, which leads to the definition of a class of Bernstein functions, which we propose to call Horn-Bernstein functions.
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Sharp inequalities related to the volume of the unit ball in $\mathbb{R}^{n}$
Xue-Feng Han,Chao-Ping Chen +1 more
TL;DR: In this article , the logarithmically complete monotonicity of a function involving the ratio of two gamma functions is presented, which yields a sharp double inequality for the quantity Ω(n+1)/γ(n + 1)/n+2.
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Sharp inequalities related to the volume of the unit ball in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{
Xue-Feng Han,Chao-Ping Chen +1 more
TL;DR: In this paper , the logarithmically complete monotonicity of a function involving the ratio of two gamma functions is presented, which yields a sharp double inequality for the quantity Ω(n+1)/γ(n + 1)/n+2.
Journal ArticleDOI
Logarithmically Completely Monotonic Rational Functions
Hamed Taghavian,Ross Drummond +1 more
TL;DR: In this paper , the class of logarithmically completely monotonic (LCM) functions is studied and conditions are proposed to ensure a rational function is LCM, a result that enables the known space of linear continuous-time externally positive systems to be enlarged and an efficient and optimal pole-placement procedure for the tracking controller synthesis problem to be developed.
References
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Book
Table of Integrals, Series, and Products
TL;DR: Combinations involving trigonometric and hyperbolic functions and power 5 Indefinite Integrals of Special Functions 6 Definite Integral Integral Functions 7.Associated Legendre Functions 8 Special Functions 9 Hypergeometric Functions 10 Vector Field Theory 11 Algebraic Inequalities 12 Integral Inequality 13 Matrices and related results 14 Determinants 15 Norms 16 Ordinary differential equations 17 Fourier, Laplace, and Mellin Transforms 18 The z-transform
A table of integrals
TL;DR: Basic Forms x n dx = 1 n + 1 x n+1 (1) 1 x dx = ln |x| (2) udv = uv − vdu (3) 1 ax + bdx = 1 a ln|ax + b| (4) Integrals of Rational Functions