A class of completely monotonic functions involving the gamma and polygamma functions

In the paper, the authors show that neither the functionnor its negative is completely monotonic on , where and are real numbers, , is the classical Euler’s gamma function, and are polygamma functions. Moreover, some other results are given in the form of remarks.


Introduction
A function f (x) is said to be completely monotonic on an interval I if it has derivatives of all orders on I and satisfies

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In the paper, the authors deny the complete monotonicity of a class of functions involving the logarithm for the classical Euler gamma function and polygamma functions. A nonnegative and infinitely differentiable funciton is called a completely monotonic function if itself and all of its derivatives change their signs alternatively. A function is completely monotonic on the positive semiaxis if and only if it is a Laplace tranform. for x ∈ I and n ≥ 0. For more information on this class of functions, please refer to Mitrinović, Pečarić, and Fink (1993, Chapter XIII) and Widder (1946, Chapter IV).
It is common knowledge that the classical Euler's gamma function is defined by for x > 0, that the logarithmic derivative of Γ(x) is called psi or digamma function and denoted by for x > 0, and that the derivatives (i) (x) for i ∈ ℕ and x > 0 are called polygamma functions.
In Alzer and Batir (2007, p. 779), Alzer and Batir obtained that (2) so is the function −G a (x) if and only if a = 0.
We remark that the function G a (x) may be reformulated as In Merkle (1998, Theorem 1), Merkle proved that the function F 0 (x) is strictly concave and the function F a (x) for a ≥ 1 2 is strictly convex on (0, ∞), where for a ≥ 0. See also Qi (2010, p. 46, Section 4.3.3).
Stimulated by Merkle (1998, Theorem 1), the authors considered in Qi (2015), its preprint Qi ( The main result of this paper is that neither which can be formulated as the following theorem. Theorem 1.1 For all real numbers a i > 0 and b i ≥ 0 and all integers i ≥ 2, neither the function f a i , b i (x) defined by (1.8) nor its negative is completely monotonic on (0, ∞).
In the final section of this paper, we also give several remarks about complete monotonicity of f a i , b i (x) and other functions related to the gamma function.

Proof of Theorem 1.1
The idea of this proof comes from the second proof of Qi (2013, Theorem 3.1) and its formally published version in Qi (2015, Theorem 2).
The famous Binet's first formula of ln Γ(x) for x > 0 is given by where for x > 0 is called the remainder of Binet's first formula for the logarithm of the gamma function Γ(x).
See Magnus (1966, p. 11) or Qi and Guo (2010, p. 462). Combining this with the integral representation for x > 0 and k ∈ ℕ = {1,2, ⋯}, see Abramowitz and Stegun (1972, p. 260, 6.4.1), yields For t > 0 and i ≥ 2, let Completely monotonic functions were characterized by Widder (1946, p. 160, Theorem 12a) which reads that a necessary and sufficient condition that f (x) should be completely monotonic in 0 ≤ x < ∞ is that where (t) is bounded and non-decreasing and the integral converges for 0 ≤ x < ∞. Therefore, in order to prove complete monotonicity of f a i , b i (x) on (0, ∞), it suffices to show the positivity or negativity of the function h i (t)−a i e −b i t on (0, ∞) , which is equivalent to Because it follows that t for i ≥ 2 and any positive number a i is neither constantly positive nor constantly negative on (0, ∞), so, by Widder (1946, p. 160, Theorem 12a) mentioned above, neither the function f a i , b i (x) for i ≥ 2, a i > 0 and b i > 0 nor its negative is completely monotonic on (0, ∞).
It is easy to calculate that for i ≥ 2. Hence, if b i = 0 and a i > 0, the function h i (t)−a i for i ≥ 2 is still neither constantly positive nor constantly negative on (0, ∞). Consequently, neither the function f a i , 0 (x) for i ≥ 2 and a i > 0 nor its negative is completely monotonic on (0, ∞). The proof of Theorem 1.1 is complete.

Remarks
In this section, we list several remarks about some functions related to the logarithm of the gamma function Γ(x). Since the function is positive and increasing on (0, ∞), see Guo and Qi (2009;2010), Zhang, Guo, and Qi (2009) and closely related references therein, the remainder (x), defined by (2.2), of Binet's first formula for the logarithm of the gamma function Γ(x) is clearly completely monotonic on (0, ∞). The formula (2.3) reveals that the functions (−1) i+1 (i) (x) are completely monotonic on (0, ∞). As a result of the facts that the sum of finitely many completely monotonic functions and the product of any positive number of completely monotonic functions are both completely monotonic, we obtain that for all real numbers a i ≤ 0 and b i ≥ 0 the function f a i , b i (x) defined by (1.8) is trivially completely monotonic on (0, ∞). in (1.7) the best possible? I believe it is.
Repeating the process in the proof of Theorem 1.1 for i = 1, we obtain that Combining these with (2.6), we see that only the function f a 1 , b 1 (x) for a 1 ≤ 1 12 and b 1 ≥ 0 is possible to be completely monotonic on (0, ∞). It is apparent that This means that, for a 1 ≤ 1 12 and b 1 ≥ 1 2 , the function f a 1 , b 1 (x) is completely monotonic on (0, ∞).
is the best possible in the sense that the scalar 1 12 cannot be replaced by a bigger one. This sharpens the main result in Qi (2013;2015), Şevli and Batir (2011).