CERTAIN INEQUALITIES INVOLVING THE Q-DEFORMED GAMMA FUNCTION

11 - The Gamma, Beta, Pi, and Psi Functions

The purpose of this paper is to study gamma functions in the context of the theory of function fields of one variable over a finite field. In particular, we explore, among other things, their connection with the theory of Drinfeld modules of rank one (which is function field cyclotomic theory), their interpolations, functional equations, the nature of their special values (transcendence, algebraicity, connection with periods) and various analogies. Most of the facts about the classical gamma function are recalled when necessary. Readers may refer to [G-K] or Artin's book on the classical gamma function. Now we describe more precisely the contents of this paper. Let Fq be a finite field of characteristic p. Section 0 covers the background material on Drinfeld modules. It also describes the gamma function F for Fq[T] (with

In this short paper, we established q- analogues of some well-known inequalities for the classical Gamma function and some inequalities bounding the ratio Γ_q (x)/Γ_q (y), where Γ_q (.) is the q-analogue of the Gamma function. The method employed in presenting the results made use of strict logarithmic convexity of a function and classical mean value theorem involving the q-analogue of the psi function. At the end, the proven result provided the q-analogue of some known result concerning the classical Gamma function.

In this paper, we introduce a new one-parameter deformation of the classical Gamma function, which we call a v-analogue. We also establish some properties generalizing those satisfied by the classical Gamma function. In addition, we establish some inequalities involving this new function.

We explore a number of functional properties of the $q$-gamma function and a class of its quotients; including the $q$-beta function. We obtain formulas for all higher logarithmic derivatives of these quotients and give precise conditions on their sign. We prove how these and other functional properties, such as the multiplication formula or the asymptotic expansion, together with the fundamental functional equation of the $q$-gamma function uniquely define those functions. We also study reciprocal "relatives" of the fundamental $q$-gamma functional equation, and prove uniqueness of solution results for them. In addition, we also use a reflection formula of Askey to derive expressions relating the classical sine function and the number $\pi$ to the $q$-gamma function. Throughout we highlight the similarities and differences between the cases $0<q<1$ and $q>1$.

In this paper, we introduce the notion of completely mixed monotonicity of a function of several variables, very few of which have appeared. We give a necessary and sufficient condition for a function constructed by ratios of gamma functions to be completely mixed monotonic. From this, some new inequalities for gamma, psi, and polygamma functions are derived.

A class of functions involving divided differences of the psi and tri-gamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving the ratio of two gamma functions and originating from the establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in <TEX>$\mathbb{R}^{n-1}$</TEX> and <TEX>$\mathbb{R}^n$</TEX> respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.

In this present paper, first, we investigate a new generalized Pochhammer's symbol and its various properties in terms of a new symbol $(s; k)$, where $s; k &gt; 0$. Then, we define a new generalization of gamma and beta functions and their various associated properties in the form of $(s; k)$. Also, we define a new generalization of hypergeometric functions and develop differential equations for generalized hypergeometric functions in the form of $(s; k)$. We present that generalized hypergeometric functions are the solution of the said differential equation. Furthermore, some useful results and properties and integral representation related to these generalized Pochhammer's symbol, gamma function, beta function, and hypergeometric functions are presented.

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. Relevant connections of the results presented here with those derived in earlier works are also pointed out.

In this paper, we prove some inequalities satisfied by the modified degenerate gamma function which was recently introduced. The tools employed include Holder’s inequality, mean value theorem, Hermite–Hadamard’s inequality, and Young’s inequality. By some parameter variations, the established results reduce to the corresponding results for the classical gamma function.

Recently, the degenerate gamma functions were introduced as a degenerate version of the usual gamma function. In this paper, we investigate several properties of these functions. Namely, we obtain an analytic continuation as a meromorphic function on the whole complex plane, the difference formula, the values at positive integers, some expressions following from the Weierstrass and Euler formulas for the ordinary gamma function, and an integral representation as an integral along a Hankel contour.

Let varGamma (x) denote the classical Euler gamma function. The logarithmic derivative psi (x)=[ln varGamma (x)]'=frac{varGamma '(x)}{ varGamma (x)}, psi '(x), and psi ''(x) are, respectively, called the digamma, trigamma, and tetragamma functions. In the paper, the authors survey some results related to the function [psi '(x)]^{2}+ psi ''(x), its q-analogs, its variants, its divided difference forms, several ratios of gamma functions, and so on. These results include the origins, positivity, inequalities, generalizations, completely monotonic degrees, (logarithmically) complete monotonicity, necessary and sufficient conditions, equivalences to inequalities for sums, applications, and the like. Finally, the authors list several remarks and pose several open problems.

In this note, a new proof for the positivity of Dunkl’s intertwining operator in thecrystallographic case is given. It is based on an asymptotic relationship between theOpdam-Cherednik kernel and the Dunkl kernel as recently observed by M. de Jeu, andon positivity results of S. Sahi for the Heckman-Opdam polynomials and their non-symmetric counterparts. 2000 AMS Subject Classification: 33C52, 33C67. 1 Introduction In [R], it was proven that Dunkl’s intertwining operator between the rational Dunkl operatorsfor a fixed finite reflection group and nonnegative multiplicity function is positive. As aconsequence we obtained an abstract Harish-Chandra type integral representation for theDunkl kernel, the image of the usual exponential kernel under the intertwiner. The proofwas based on methods from the theory of operator semigroups and a rank-one reduction.In the present note, we give a new, completely different proof of these results under theonly additional assumption that the underlying reflection group has to be crystallographic.In contrast to the proof of [R], where precise information on the supports of the representingmeasures could only be obtained by going back to estimates of the kernel from [J1], thisinformation is now directly obtained. Our new approach relies first on an asymptotic rela-tionship between the Opdam-Cherednik kernel and the Dunkl kernel as recently observed byde Jeu [J2], and second on positivity results of Sahi [S] for the Heckman-Opdam polynomialsand their non-symmetric counterparts.

In this paper, some convexity properties and some inequalities forthe (p; k)-analogue of the Gamma function,a (p; k)-analogue of the celebrated Bohr-Mollerup theorem is given. Furthermore, a (p; k)-analogue of the Riemann zeta function,p;k(x)is introduced andsome associated inequalities are derived. The established results provide the(p; k)-generalizations of some known results concerning the classical Gammafunction

Abstract. In this paper, the authors establish some double in-equalities concerning the ( q;k )-deformed Gamma function. Theseinequalities emanate from certain problems of tra–c °ow. Theprocedure makes use of the integral representation of the ( q;k )-deformed Gamma function. 1. IntroductionThe celebrated classical Euler’s Gamma function, i( x ) is usually deflnedfor x > 0 byi( x ) =Z 1 0 t xi 1 e it dt = lim n!1 • n ! n x x ( x +1) ::: ( x + n )‚ : The k -deformed Gamma Function, i k ( x ) (also known as the k -analogueof the Gamma function or simply the k -Gamma function) is deflned by(see [3])i k ( x ) =Z 1 0 e i tkk t xi 1 dt; k > 0 ; x > 0 : It satisfles the following properties (see [3]).i k ( x + k ) = x i k ( x ) ; i k ( k ) = 1 : Received March 20, 2015. Accepted December 22, 2015.2010 Mathematics Subject Classiflcation. 33B15, 33D05.Key words and phrases. Gamma function, q -deformed Gamma function, k -deformed Gamma function, ( q;k )-deformed Gamma function, q -integral, Inequality.*Corresponding author