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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The Australian Journal of Mathematical Analysis and Applications
TL;DR: In this paper, a fixed point method for proving the Hyers-Ulam stability of the functional equation f(x + y) = f (x)f(y) f( x + y + f(y)) f(X + y)+ f(Y)+f(Y).
Journal ArticleDOI
From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions
Feng Qi,Feng Qi,Bai-Ni Guo +2 more
TL;DR: In this paper, the authors review origins, motivations, and generalizations of a series of inequalities involving finitely many exponential functions and sums, and present complete monotonicity of a linear combination of Finite Many Gamma Functions.
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Infinitely Log-monotonic Combinatorial Sequences
TL;DR: By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, it is shown that the sequences of the Bernoulli numbers, the Catalan numbers and the central binomial coefficients are infinitely Log-Monotonic.
Journal ArticleDOI
More supplements to a class of logarithmically completely monotonic functions associated with the gamma function
Senlin Guo,Feng Qi +1 more
TL;DR: In this article, a necessary and sufficient condition and a neces- sary condition are established for a function involving the gamma function to be logarithmically completely monotonic on (0, 1).
Journal ArticleDOI
Some inequalities and monotonicity properties associated with the gamma and psi functions and the Barnes G-function
TL;DR: In this paper, several properties associated with inequalities and the logarithmically complete monotonicity of functions related to the gamma and psi functions and the Barnes G-function are obtained.
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