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The monotonicity and convexity of a function involving psi function with applications [PDF]
Journal of Inequalities and Applications, 2016 In this paper, we prove that the function x ↦ exp ( ψ ( x + 1 2 ) − 1 24 1 x 2 + 7 / 40 ) − x $$ x\mapsto\exp \biggl( \psi \biggl( x+\frac{1}{2} \biggr) -\frac {1}{24} \frac{1}{x^{2}+7/40} \biggr) -x $$ is decreasing from ( − 1 / 2 , ∞ ) $( -1/2,\infty )
Bang-Cheng Sun +3 more
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The quotient of gamma functions by the psi function [PDF]
Computational & Applied Mathematics, 2011 The aim of this paper is to construct the asymptotic series of the ratio of gamma functions by Kershaw then to deduce some sharp estimates. Mathematical subject classification: 33B15, 05A16, 26D15.
Cristinel Mortici
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On some inequalities for the gamma and psi functions [PDF]
Mathematics of Computation, 1997 Summary: We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, star-shaped, and super-additive functions which are related to \(\Gamma\) and \(\psi\).
Horst Alzer
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Monotonicity of some functions involving the gamma and psi functions [PDF]
Mathematics of Computation, 2008 Let L(x) := x-Γ(x+t)/Γ(x+s) x 8-t+1 where Γ(x) is Euler's gamma function. We determine conditions for the numbers s, t so that the function Φ(x):= Γ(x+s)/Γ(x+t)x t-s-1 L"(x) is strongly completely monotonic on (0, oo). Through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context
Stamatis Koumandos
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Some completely monotonic functions related to the psi function [PDF]
Mathematical Inequalities & Applications, 2011 Complete monotonicity properties of some functions involving the psi function are studied and some known results are extended and generalized. Moreover, a necessary and sufficient conditions for some functions to be completely monotonic are presented and proved.
Tomislav Burić, Neven Elezović
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Chebyshev Approximations for the Psi Function [PDF]
Mathematics of Computation, 1973 Rational Chebyshev approximations to the psi (digamma) function are presented for .5 ≦ x ≦ 3.0 .5 \leqq x \leqq 3.0 , and 3.0 ≦ x 3.0 \leqq x . Maximum relative errors range down to the order of 10 − 20
W. J. Cody +2 more
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ON FUNCTIONAL INEQUALITIES FORTHE PSI FUNCTION
Issues of AnalysisIn this note we study the monotonicity of the function $x\mapsto (1 +bx)^a/ (1 + ax)^b$. We also give the several inequalities involving the psi function, whic is the logarithmic derivative of the gamma function.
Barkat Ali Bhayo, Lixin Xie, Sumeyye Yar
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The Laplace transform of the psi function [PDF]
Proceedings of the American Mathematical Society, 2009 An expression for the Laplace transform of the psi function L(a) := ∫ ∞ 0 e -at ψ(t+1)dt is derived using two different methods. It is then applied to evaluate the definite integral M(a) = 4 / π ∫ ∞ 0 x 2 dx/ x2+ln 2 (2e -a cos x) , for a > In 2 and to resolve a conjecture posed by Olivier Oloa.
Atul Dixit
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On values of the psi function [PDF]
Journal of Applied Mathematics and Computational Mechanics, 2017 Marcin Adam +4 more
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Some properties of the gamma and psi functions, with applications [PDF]
Mathematics of Computation, 2004 In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some well-known results for the volume Ω n \Omega _n of the unit ball B
S.-L. Qiu, Матти Вуоринен
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