Some complete monotonicity properties for the -gamma function
Introduction
The familiar (Euler) gamma function is defined (for ) by
The digamma (or psi-) function is defined (for ) as the logarithmic derivative of Euler’s gamma function , that is, by
The following integral and series representations are known (see [1], [4], [18], [23]):where γ denotes the Euler–Mascheroni constant defined by (see also a recent work [7])
Euler also introduced the following interesting variant of the gamma function (see [3], [21]):so that
The p-analogue of the ψ-function is defined as the logarithmic derivative of the -function as follows (see [16]):
The following representations for the functions and hold true:and
Jackson [9], [10], [11] (see also [23], [25]) defined the basic (or q-) analogue of the gamma function as follows:andwhere, and in what follows,and
The basic (or q-) gamma function has the following integral representation (see, for details, [9], [10], [11]):where defined byis the q-analogue of the classical exponential function . The q-analogue of the ψ-function is defined (for ) as the logarithmic derivative of the q-gamma function, that is, by (see also [17])Many properties of the q-gamma function were derived by Askey [1], [4] (see also [23, p. 490 et seq.]).
It is well-known that
For and , we find from (1.9) thatAlso, for and , we find from (1.10) that
A Stieltjes integral representation for is given in [8]. It is well-known that the function is strictly completely monotonic on , that is, we have [[1], [4]Thus, from (1.11), (1.12), we conclude that has the same property for any , that is, thatLet . Then, by using the second representation of given in (1.11), we can show thatwhich yieldsIf , then we find from the second representation of given in (1.12) thatandHence we have Definition 1 For and for , the -gamma function is defined bywhere It is easy to see from the definition (1.15) that the function fits into the following commutative diagrams: We define -analogue of the ψ-function as the logarithmic derivative of the -gamma function, that is, by A function f is called log-convex if, for all such that and for all , the following inequality holds true:or, equivalently, ifWe now give some definitions about completely monotonic functions. Definition 2 A function f is said to be completely monotonic on an open interval I if f has derivatives of all orders on I and satisfies the following inequalities:If the inequality (1.17) is strict, then the function f is said to be strictly completely monotonic on I. Definition 3 A positive function f is said to be logarithmically completely monotonic (see [20]) on an open interval I, if f satisfies the following inequalities:If the inequality (1.18) is strict, then the function f is said to be strictly logarithmically completely monotonic.
Let C and L denote the set of completely monotonic functions and the set of logarithmically completely monotonic functions, respectively. The relationship between completely monotonic functions and logarithmically completely monotonic functions can be presented by (see [2]).
The following theorem gives an integral characterization of completely monotonic function: Theorem 1 Hausdorff–Bernstein–Widder Theorem A functionis completely monotonic on if and only if it is the Laplace transform of a finite non-negative Borel measure μ on , that is, if the function φ is of the following form: Remark 1 Some properties of completely monotonic function are as follows: A non-negative finite linear combination of completely monotonic functions is completely monotonic. The product of two completely monotonic functions is completely monotonic.
Section snippets
A set of main results
We start with the following lemma. Lemma 1 Let such that . Then Proof By applying Young’s inequality:we have Now, in order to prove thatby applying Young’s inequality (2.1) once again, we getwhich obviously completes the proof of Lemma 1. □ Theorem 2 The functionis
Logarithmically completely monotonic functions
Theorem 3 Let such thatand Then the function given byis completely monotonic on . Proof We first define a function by Then, for , we have Alzer [2] showed that, if a
Application for the function
In this concluding section, we give extensions of several results given by Shabani [22] to hold true for the function . Since the proofs are almost similar, we choose to omit them here (see also [12], [13], [15]). Lemma 3 Let and e be real numbers such thatThen Lemma 4 Let and be real numbers such that If or , then Lemma 5 Let and be real numbers
Remarks and observations
Various families of basic (or q-) series and basic (or q-) polynomials, which do fairly frequently involve the basic (or q-) gamma function , the basic (or q-) beta function and the basic (or q-) digamma or the ψ-function , are useful in a wide variety of fields including, for example, theory of partitions, number theory, combinatorial analysis, finite vector space, Lie theory, particle physics, non-linear electric circuit theory, mechanical engineering, theory of heat
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2014, Applied Mathematics and ComputationErratum: Some complete monotonicity properties for the (p, q)-gamma function (Applied Mathematics and Computation (2014) 227 (662))
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2016, FilomatSome conditions for a class of functions to be completely monotonic
2015, Journal of Inequalities and Applications