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 new lower bound in the second Kershaw's double inequality
TL;DR: In this article, a new and elegant lower bound in the second Kershaw's double inequality is established, some alternative simple and polished proofs are given, several deduced functions involving the gamma and psi functions are proved to be decreasingly monotonic and logarithmically completely monotone.
Journal ArticleDOI
Some logarithmically completely monotonic functions related to the gamma function
Feng Qi,Bai-Ni Guo +1 more
TL;DR: In this article, the logarithmically complete monotonicity of some functions such as 1 for fi 2 R on (i1;1) or (0,1) is obtained, some known results are recovered, extended and generalized.
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Complete monotonicity of a function involving the divided difference of digamma functions
Feng Qi,Qiu-Ming Luo,Bai-Ni Guo +2 more
TL;DR: In this article, necessary and sufficient conditions are provided for a function involving the divided difference of two psi functions to be completely monotonic, and a class of inequalities for sums are presented, and two double inequalities for bounding the ratio of two gamma functions are discovered.
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Complete monotonicity of functions involving the $q$-trigamma and $q$-tetragamma functions
TL;DR: In this paper, it was shown that the monotonic properties of four functions involving the $q$-digamma function and two double inequalities for bounding the q$-trigamma functions are monotone.
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Complete Monotonicity of Logarithmic Mean
TL;DR: In this article, the logarithmic mean is proved to be completely monotonically monotonic and an open problem about the complete monotonicity of the extended mean values is posed.
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