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 The logarithmic derivative of is called the psi or digamma function. That is, where is the Euler–Mascheroni constant, and for 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 and as where . 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]): and

Very recently, Nantomah, Prempeh and Twum [35] introduced a -analogue of the gamma and digamma functions defined for , and as and It is obvious that . Some important identities and inequalities involving these functions may be found in [30, 34, 35].

In [4], the function was proved to be strictly increasing on . In [6], it is demonstrated that if and , then Furthermore, Guo and Qi [14] showed that the function is strictly increasing and concave on . Attracted by this work, it is natural to look for an extension of (1.7) involving and . 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 , 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.

Theorem 1.1

For , the function is strictly increasing on . In particular, the inequalities hold true for and where the constants and 0 in (1.8) are the best possible.

Remark 1.1

Here, we give an application of Theorem 1.1. Define the k-generalization of the Nielsen’s β-function as By using (1.8), we easily obtain double inequalities of the generalized Nielsen’s β-function for and :

Theorem 1.2

For , the function is strictly concave on . As a result, for and , we have

Using the Theorems 1.1 and 1.2, we easily obtain the following Corollary 1.1.

Corollary 1.1

For and , we have and

Theorem 1.3

For and , we have

Theorem 1.4

For and every positive integer , the function is strictly decreasing on with and As a result, for and every positive integer , we have

Theorem 1.5

For , the inequalities hold where .

Lemmas

Lemma 2.1

[42] If f is a function defined in an infinite interval I such that for some , then on I.

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.

Lemma 2.2

For , the function is positive on if and only if .

Proof

Direct computation yields and It is easily observed that if and only if . We complete the proof by using Lemma 2.1. □

Lemma 2.3

The following limit identity holds true:

By applying twice l’Hôspital rule, we easily complete the proof. □

Lemma 2.4

For , the inequalities hold true for any .

Using the inequalities in [34], namely we easily obtain (2.2) as . □

Lemma 2.5

([25, 26, 43, 48])

Let and () be real numbers such that and be increasing (resp., decreasing), then is increasing (resp., decreasing).

Lemma 2.6

For and every positive integer , the following limit identity holds true:

Considering the inequalities (see [34, Theorem 2.7]) and differentiating them times, we easily complete the proof. □

Proofs of theorems

First proof of Theorem 1.1

A simple calculation gives and Using Lemma 2.2, we easily obtain This implies that the function is strictly increasing, and so on . As a result, the function is also strictly increasing on . 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 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 , we easily get This implies that the function is strictly decreasing on with . Hence, we have . The proof is completed. □

Third proof of Theorem 1.1

Direct calculation results in and with .

In order to prove for , it suffices to show So, we only need to prove which is valid. By using Lemma 2.1, we can conclude that . Hence, the function is strictly increasing on . □

Proof of Theorem 1.2

Using formula (3.7), we have For , the fact is equivalent to Applying inequality (3.8), we need to prove An easy calculation yields and with , where and For , we easily obtain This implies that is strictly decreasing and is strictly increasing on . Using and Lemma 2.1, we complete the proof. □

Proof of Theorem 1.3

Using (1.1) and (1.2), we get By the mean value theorem for differentiation, there exists a number such that and Hence, we find It is well known that the function is strictly increasing in k on with Therefore, we get This completes the proof. □

Proof of Theorem 1.4

By (1.6) and direct computation, we have where . Let us define sequences , and by and It follows that It is not difficult to see that the fact is equivalent to So the sequence is strictly decreasing. This implies that the function is strictly decreasing on by Lemma 2.5. From the identity we easily obtain (1.14). Using Lemma 2.6, we get (1.13). This completes the proof. □

Proof of Theorem 1.5

Using (1.4) and the functional equation (see [35]) we obtain, after a direct computation, that and Combining (3.12) and (3.13) with (3.14), we get By the mean value theorem, we obtain Hence, identity (3.15) changes into From identity (3.16), we conclude that Next, we show that ρ is strictly increasing on . Differentiating , we observe that if and only if which follows from the geometric–logarithmic mean inequality. A simple computation yields and . Since and are strictly increasing on , we easily obtain that Hence we have Replacing x by here completes the proof. □

A conjecture

Finally, we give a conjecture.

Conjecture 4.1

For and , the function is strictly decreasing from onto .

Remark 4.1

It is natural to ask whether the monotonicity result of Theorem 1.1 can be extended to the digamma function with two parameters by using the method of Theorem 1.1. Unfortunately, we failed to prove Conjecture 4.1. Alzer’s work shows that the function is useful for studying harmonic numbers. This is related to the formula (see [35, Remark 2.1]) where is the nth harmonic number. So, it would be a meaningful result if anyone can prove this conjecture.

Remark 4.2

The -generalized Nielsen’s β-function can be defined as where , , and . Analogously to Remark 1.1, if Conjecture 4.1 holds true, we can estimate the upper and lower bounds of this function .

Results and discussion

Some monotonicity and concavity properties of the k and -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 . 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 -analogue of the digamma function.

Methods and experiment

Not applicable.