Inequalities and monotonicity properties for the gamma function (original) (raw)
Monotonicity results for the gamma function
2003
is strictly increasing on [1, ∞), respectively. From these, some inequalities, for example, the Minc-Sathre inequality, are deduced, and two open problems posed by the second author are solved partially.
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.
Further Inequalities Associated with the Classical Gamma Function
Turkish Journal of Analysis and Number Theory, 2016
In this paper, we present some double inequalities involving certain ratios of the Gamma function. These results are further generalizations of several previous results. The approach is based on the monotonicity properties of some functions involving the the generalized Gamma functions. At the end, we pose some open problems.
A complete monotonicity property of the gamma function
Journal of Mathematical Analysis and Applications, 2004
A logarithmically completely monotonic function is completely monotonic. The function is strictly completely monotonic on (0,∞). The function is strictly logarithmically completely monotonic on (0,∞).
Inequalities and monotonicity properties for the gamma functions
2001
We study the variation of the zeros of the Hermite function H\(t) with respect to the positive real variable A. We show that, for each non-negative integer n, H\(t) has exactly n + 1 real zeros when n < A ^ n + 1, and that each zero increases from-oo to oo as A increases. We establish a formula for the derivative of a zero with respect to the parameter A; this derivative is a completely monotonic function of A. By-products include some results on the regular sign behaviour of differences of zeros of Hermite polynomials as well as a proof of some inequalities, related to work of W. K. Hayman and E. L. Ortiz for the largest zero of H\(t). Similar results on zeros of certain confluent hypergeometric functions are given too. These specialize to results on the first, second, etc., positive zeros of Hermite polynomials.
A generalization of some inequalities for the gamma function
Journal of Computational and Applied Mathematics, 1998
Laforgia (1984) obtained some inequalities of the type F(k + 2) 1 F(k + 2) 1-> or-< F(k + 1) (k + ~)1-~ F(k + 1) (k + ~z) 1-a' according to the values of the positive parameters ~ and 2, valid for every non-negative real value of k, or at least for k greater than or equal than a k o depending on a and 2. In this paper a complete analysis of the problem is carried out, in order to establish, for fixed 0¢ and 2, which of the two former inequalities holds, and for which values ofk. ~. t-E3 1997 Elsevier Science B.V. All rights reserved.
Monotonicity Results and Associated Inequalities for K-Gamma Function
Anadolu University Journal of Science and Technology-A Applied Sciences and Engineering
This study is inspired by the work of Neumann in 2011. In the study, we establish some double inequalities involving the ratio (+) (+) , where is the-analogue of Euler's gamma function. Some monotonicity results involving k-gamma function are found. By the aid of these results, some inequalities such as [Γ (+)] ≤ ∑ [Γ (+)] =1 for > 0, 1 ≤ ≤ , where = 1 + 2 + ⋯ + are valid.
A refinement of a double inequality for the gamma function
2010
In the paper, we present a monotonicity result of a function involving the gamma function and the logarithmic function, refine a double inequality for the gamma function, and improve some known results for bounding the gamma function.
Some monotonicity properties and inequalities for the (p, k)-gamma function
Kragujevac Journal of Mathematics, 2018
In this paper, the authors present some complete monotonicity properties and some inequalities involving the (p, k)-analogue of the Gamma function. The established results provide the (p, k)-generalizations for some results known in the literature. Γ p,k (k) = 1.
Inequalities and monotonicity properties for gamma and q-gamma functions
arXiv (Cornell University), 2013
Most of the errors in the original paper had to do with saying that certain functions related to the q-gamma function were not completely monotonic. We discovered these errors through reading the paper Some completely monotonic functions involving the q-gamma function, by Peng Gao, http://arxiv.org/abs/1011.3303\. We also take the opportunity to correct some errors in other places including the statement and proof of Theorem 3.4.
On Some Inequalities for the Gamma Function
2013
We present some elementary proofs of well-known inequalities for the gamma function and for the ratio of two gamma functions. The paper is purely expository and it is based on the talk that the first author gave during the memorial conference in Patras, 2012.
Applied Mathematics and Computation, 2008
In this article, the authors present a necessary condition, a sufficient condition and a necessary and sufficient condition for a class of functions associated with the gamma function to be logarithmically completely monotonic. As a consequence of these results, supplements to the recent investigation by Chen and Qi [J. Math. Anal. Appl. 321 (2006), 405-411] are provided and a new Kečkić-Vasić type inequality is concluded.
Extensions of Some Inequalities for the Gamma Function
In this paper, we improve the results of Shabani [7] concerning some inequalities for the Gamma function. Our approach makes use of the logarithmic derivative of products of the Gamma function. We also present some p-analogues.
Further inequalities for the gamma function
Mathematics of Computation, 1984
For X > 0 and k > 0 we present a method which permits us to obtain inequalities of the type (it + a)x_l < T(k + \)/T(k + 1) < (k + ß)x'x, with the usual notation for the gamma function, where a and ß are independent of k. Some examples are also given which improve well-known inequalities. Finally, we are also able to show in some cases that the values a and ß in the inequalities that we obtain cannot be improved.