A complete monotonicity property of the gamma function (original) (raw)
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In this article, a necessary condition, several sufficient conditions and a sufficient and necessary condition for a class of functions involving the gamma function to be logarithmically completely monotonic are established, and then some known results are extended.
Complete Monotonicity and Inequalities of Functions Involving Gamma\GammaGamma-function
arXiv (Cornell University), 2017
In this paper, we investigate the complete monotonicity of some functions involving gamma function. Using the monotonic properties of these functions, we derived some inequalities involving gamma and beta functions. Such inequalities arising in probability theory. Due to the relation between gamma and beta function, we derived some inequalities that are involving beta function. Commonly, beta function is defined by B(x, y) = 1 0 t x−1 (1 − t) y−1 dt, x and y > 0, and its relation with the gamma function is given by B(x, y) = Γ(x + y) Γ(x)Γ(y) , x, y > 0. The layout of the paper: In the first section, we prove our main results. In the second section, we apply some of our main results to prove the inequality Γ(x + 1)Γ(x − a − b + 1) Γ(x − a + 1)Γ(x − b + 1) Γ(y − a + 1)Γ(y − b + 1) Γ(y + 1)Γ(y − a − b + 1) ≥ 1, y ≥ x ≥ a + b > b ≥ a > 0. The last section is devoted for the concluding remarks. 2 The Main Results In this section, we present and prove our main results. First, we present some useful definitions and theorems. Definition 2.1. A function f (z) is called completely monotonic on an interval I if it has derivatives of any order f (n) (z), n = 0, 1, 2, 3, • • •, and if (−1) n f (n) (z) ≥ 0 for all x ∈ I and all n ≥ 0. If the above inequality is strict for all x ∈ I and all n ≥ 0, then f (z) is called strictly completely monotonic. Definition 2.2. A function f (z) is called logarithmically completely monotonic on an interval I if its logarithm has derivatives [ln f (z)] (n) of orders n ≥ 1, and if (−1) n [ln f (z)] (n) ≥ 0 for all x ∈ I and all n ≥ 1. If the above inequality is strict for all x ∈ I and all n ≥ 1, then f (z) is called strictly logarithmically completely monotonic. Theorem 2.1. [29]. Every (strict) logarithmically completely monotonic function is (strict) completely monotonic. Now, we turn to prove our main results. Theorem 2.2. Let a, b ≥ 0. Define f (z) = Γ(z + 1)Γ(z − a − b + 1) Γ(z − a + 1)Γ(z − b + 1) , z > a + b − 1. Then f (z) is completely monotonic function.
A result regarding monotonicity of the Gamma function
Acta Universitatis Sapientiae, Mathematica, 2017
In this paper we analyze the monotony of the function ln Γ(x)ln (x2+τ)-ln (x+τ) rmln,Gamma(rmx)overrmln,(rmx2+tau)−rmln,(rmx+tau){{{\rm{ln}}\,\Gamma ({\rm{x}})} \over {{\rm{ln}}\,({\rm{x}}^2 + \tau) - {\rm{ln}}\,({\rm{x}} + \tau)}}rmln,Gamma(rmx)overrmln,(rmx2+tau)−rmln,(rmx+tau) , for τ > 0. Such functions have been used from different authors to obtain inequalities concerning the gamma function.