Some monotonicity properties and inequalities for the generalized digamma and polygamma functions

Several monotonicity and concavity results related to the generalized digamma and polygamma functions are presented. This extends and generalizes the main results of Qi and Guo and others.


Introduction
The Euler gamma function is defined for all positive real numbers x by (x) = ∞ 0 t x-1 e -t dt.
The logarithmic derivative of (x) is called the psi or digamma function. That is, where γ = 0.5772 . . . is the Euler-Mascheroni constant, and ψ (m) (x) for m ∈ N are known as the polygamma functions. The gamma, digamma and polygamma functions play an important role in the theory of special functions, and have many applications in other many branches, such as statistics, fractional differential equations, mathematical physics and theory of infinite series. The reader may see the references [9-13, 18-20, 24, 45-47, 49]. Some of the work on the complete monotonicity, convexity and concavity, and inequalities of these special functions can be found in [1-6, 8, 14-17, 21, 22, 27-30, 37-42] and the references therein. In 2007, Diaz and Pariguan [11] defined the k-analogue of the gamma function for k > 0 and x > 0 as k (x) = ∞ 0 t x-1 e -t k k dt = lim n→∞ n!k n (nk) x k -1 x(x + k) · · · (x + (n -1)k) , where lim k→1 k (x) = (x). Similarly, we may define the k-analogue of the digamma and polygamma functions as It is well known that the k-analogues of the digamma and polygamma functions satisfy the following recursive formula and series identities (see [11]): Very recently, Nantomah, Prempeh and Twum [35] introduced a (p, k)-analogue of the gamma and digamma functions defined for p ∈ N, k > 0 and x > 0 as and It is obvious that lim p→+∞ ψ p,k (x) = ψ k (x). Some important identities and inequalities involving these functions may be found in [30,34,35].
In [4], the function φ(x) = ψ(x) + ln(e 1 x -1) was proved to be strictly increasing on (0, ∞). In [6], it is demonstrated that if a ≤ -γ and b ≥ 0, then (1.7) Furthermore, Guo and Qi [14] showed that the function φ(x) is strictly increasing and concave on (0, ∞). Attracted by this work, it is natural to look for an extension of (1.7) involving ψ k (x) and ψ p,k (x). On the other hand, Nielsen's β-function has been deeply researched in the last years. In particular, K. Nantomah gave some results on convexity and monotonicity of the function in [31], and obtained some convexity and monotonicity results as well as inequalities involving a generalized form of the Wallis's cosine formula in [32]. The function can be used to calculate some integrals (see [7,36]). Recently, K. Nantomah studied the properties and inequalities of a p-generalization of the Nielsen's function in [33]. In this paper, we shall give double inequalities for the k-generalization of the Nielsen β-function. In addition, it is worth noting that Krasniqi, Mansour, and Shabani presented some inequalities for q-polygamma functions and q-Riemann Zeta functions by using a q-analogue of Hölder type inequality in [23]. The first aim of this paper is to present a new monotonicity theorem for ψ k (x), and give three different proofs. The second aim is to show an inequality for the ratio of the generalized polygamma functions by generalizing a method of Mehrez and Sitnik. The classical Mehrez and Sitnik's method may be found in [25,26,43]. Finally, we also give a new inequality for the inverse of the generalized digamma function.
Our main results read as follows.

Lemmas
Lemma 2.1 [42] If f is a function defined in an infinite interval I such that Remark 2.1 Lemma 2.1 was first proposed by Professor Feng Qi. It is simple, but has been validated in [15,41,42] to be especially effective in proving monotonicity and complete monotonicity of functions involving the gamma, psi and polygamma functions. The reader may refer to [40] and the references therein.
Proof Direct computation yields It is easily observed that β(x + k)β(x) < 0 if and only if k ≤ 1. We complete the proof by using Lemma 2.1.

Lemma 2.3
The following limit identity holds true: Proof By applying twice l'Hôspital rule, we easily complete the proof.

Proofs of theorems
First proof of Theorem 1.1 A simple calculation gives Using Lemma 2.2, we easily obtain This implies that the function μ k (x) is strictly increasing, and so δ k (x) > 0 on (0, ∞). As a result, the function e φ k (x) is also strictly increasing on (0, ∞). Considering Lemma 2.3, we have The proof of Theorem 1.1 is completed.
Second proof of Theorem 1.1 It is easily observed that δ k (x) > 0 is equivalent to Considering Lemma 2.4, we only need to prove Taking the logarithm to both sides of (3.2), we prove So, we only need to prove Since k ≤ 1, we easily get This implies that the function λ k (x) is strictly decreasing on (0, ∞) with lim x→∞ λ k (x) = 0. Hence, we have λ k (x) > 0. The proof is completed.

Third proof of Theorem 1.1 Direct calculation results in
and with lim x→+∞ φ k (x) = 0.
In order to prove φ k (x)φ k (x + k) > 0 for x > 0, it suffices to show So, we only need to prove (3.9) which is valid. By using Lemma 2.1, we can conclude that φ k (x) > 0. Hence, the function φ k (x) is strictly increasing on (0, ∞).
This implies that k (x) is strictly decreasing and k (x) is strictly increasing on (0, ∞).
Proof of Theorem 1.3 Using (1.1) and (1.2), we get By the mean value theorem for differentiation, there exists a number σ k,n = σ k,n (x) such that 0 < σ k,n < x and Hence, we find It is well known that the function σ k,n is strictly increasing in k on [1, +∞) with Therefore, we get This completes the proof.
Proof of Theorem 1.4 By (1.6) and direct computation, we have and It follows that It is not difficult to see that the fact So the sequence {ω m,i } i≥0 is strictly decreasing. This implies that the function φ m,p,k (x) is strictly decreasing on (0, ∞) by Lemma 2.5. From the identity we easily obtain (1.14). Using Lemma 2.6, we get (1.13). This completes the proof.

Results and discussion
Some monotonicity and concavity properties of the k and (p, k)-analogues of the digamma and polygamma functions were deeply studied. In doing so, we established some inequalities involving the generalized digamma and polygamma functions. Theorems 1.1-1.3 are extensions of some known results. Theorem 1.4 is not only a completely new result, it's even new for ψ(x). In addition, the method of proof is also new. Theorem 1.5 gives an inequality for the inverse of the digamma function. At the moment, such results are very few.
In the end, we stated a conjecture involving the (p, k)-analogue of the digamma function.

Methods and experiment
Not applicable.