Abstract

We prove some monotonicity properties of functions involving gamma and q-gamma functions.

1. Introduction

The -gamma function is defined for positive real numbers and by We note here according to [1] that the limit of as gives back the well-known Euler's gamma function: Note also that from the definition we have for positive and , we see that . For historical remarks on gamma and -gamma functions, we refer the reader to [1–3].

There exists an extensive and rich literature on inequalities for the gamma and -gamma functions. For some developments in this area, we refer the reader to the articles in [2–7] and the references therein. Many of these inequalities follow from the monotonicity properties of functions which are closely related to (resp., ) and its logarithmic derivative (resp. ). Here we recall that a function is said to be completely monotonic on if it has derivatives of all orders and , , and is said to be strictly completely monotonic on if , , . Lemma 2.3 below asserts that is completely monotonic on an interval if is. Following [8], we call such functions logarithmically completely monotonic. For similar notions of logarithmically completely monotonic functions and the history of the notions, we refer the reader to the articles in [9–12].

We note here that (see [13]) and that and are completely monotonic functions on (see [5, 14]). Thus, one expects to deduce results on gamma and -gamma functions from properties of (logarithmically) completely monotonic functions, by applying them to functions related to or . It is our goal in this paper to obtain some results on gamma and -gamma functions via this approach. As an example, we recall the following result of Bustoz and Ismail (the case ) as well as Ismail and Muldoon.

Theorem 1.1 (see [15, Theorem  3], [4, Theorem  2.5]). Let , and for and , define For , let . Then is logarithmically completely monotonic on if and is logarithmically completely monotonic on if .

It follows immediately from the above theorem that for and , one has [4] for : Alzer later [3] determined the best values such that the inequalities hold for to be

Motivated by the above results, we will show in Section 3 that the function as defined in Theorem 1.1 is logarithmically completely monotonic on for . This will enable us to deduce the left-hand side inequality of Alzer's result above for . The derivatives of are known as polygamma functions and in Section 4 we will prove some inequalities involving the polygamma functions.

2. Lemmas

The following lemma lists some facts about and . These can be found, for example, in [3, equation (2.7)] and [5, equations  (1.2)–(1.5)].

Lemma 2.1. For ,

The set of all the completely monotonic functions on an interval is equipped with a ring structure with the usual addition and multiplication of functions. The next two simple lemmas can also be used to construct new (logarithmically) completely monotonic functions; part of Lemma 2.2 is contained in [4, Lemma  1.3].

Lemma 2.2. If is completely monotonic on some interval , then so is on for any . Consequently, if is logarithmically completely monotonic on some interval , then so is on for any .

Lemma 2.3 (see [15, Lemma  2.1]). If is completely monotonic on an interval, then is also completely monotonic on the same interval.

Lemma 2.4. Let and be real numbers such that , , and for . If the function is decreasing and convex on , then If , then one only needs to be convex for the above inequality to hold.

The above lemma, except the last part, is a special case of Lemma  2 in [2]. This lemma follows from the theory of majorization, for example, see the discussions in [16].

Lemma 2.5 (Hadamard's inequality). Let be a convex function on , then

Lemma 2.6. For real numbers and any integer ,

Proof. We can recast the above inequality as where for positive numbers and real numbers with , we define The assertion of the lemma now follows from the fact that is increasing in both and and in both and (see, e.g., Theorem  1 in [17] for a proof of this).

Lemma 2.7. Let be two integers, then for any fixed constant , the function has exactly one root when .

Proof. We have The function is clearly decreasing when . By considering the cases and we conclude that has exactly one root when . It follows from this and Cauchy's mean value theorem that has at most two roots when . This combined with the observation that and yields the desired conclusion.

Lemma 2.8. For , one has

Proof. We write and we observe that where the last inequality follows from The above inequality follows from the fact that the derivative of is for and that . We now deduce that for . This implies that the function is increasing for . Thus we get which is the desired result.

3. Main Results

Theorem 3.1. Let and be real numbers such that , , and for . If is completely monotonic on , then is logarithmically completely monotonic on .

Proof. It suffices to show that is completely monotonic on or for , By Lemma 2.4, it suffices to show that is decreasing and convex on or equivalently, and for . The last two inequalities hold since we assume that is completely monotonic on . This completes the proof.

As a direct consequence of Theorem 3.1, we now generalize a result of Alzer [2, Theorem  10].

Corollary 3.2. Let and   be real numbers such that , , and for . Then, is logarithmically completely monotonic on .

Proof. Apply Theorem 3.1 to and note that is completely monotonic on and this completes the proof.

Theorem 3.3. Let be completely monotonic on , then for , the functions are logarithmically completely monotonic on .

Proof. We may assume . We will prove the first assertion and the second one can be shown similarly. It suffices to show that is completely monotonic on or for , The last inequality holds by Lemma 2.5 and our assumption that is completely monotonic on . This completes the proof.

Corollary 3.4. For , the functions are logarithmically completely monotonic on .

Proof. Apply Theorem 3.3 to and note that is completely monotonic on and this completes the proof.

By applying Lemma 2.5 to , we obtain the following theorem.

Theorem 3.5. For positive and ,

The upper bound in Theorem 3.5 is due to Ismail and Muldoon [4]. Our proof here is similar to that of Corollary  3 in [18].

Our next result refines the left-hand side inequality of (1.6) for .

Theorem 3.6. Let and . Let be defined by (1.7) and let the function be defined as in Theorem 1.1. Then is logarithmically completely monotonic on .

Proof. Define It suffices to show that is completely monotonic on . We have Using the expression (2.1), we can rewrite as Expanding , we may further rewrite as where In order for to be completely monotonic on , it suffices to show for . This is just Lemma 2.6 and this completes the proof.

Theorem 3.6 implies that , where the limit can be easily evaluated using (1.1) and we recover the left-hand side inequality of (1.6) for .

For , let Alzer [19, Theorem  1] showed that for fixed , the function is strictly completely monotonic on if and only if and the function is strictly completely monotonic on if and only if . In view of Lemmas  2.2 and  2.3, Alzer's result would follow if one can show that is strictly completely monotonic on if and only if and that is strictly completely monotonic on if and only if . We now establish the above assertions in the following.

Proposition 3.7. For , let be defined as in (3.15). Then is strictly completely monotonic on if and only if and is strictly completely monotonic on if and only if .

Proof. As is completely monotonic on , we may just focus on the case or 0. We have Using the asymptotic expressions (2.4) and (2.5), we see that for any integer . It is then easy to see that the “if” part of the assertions of the proposition will follow if we can show that is strictly completely monotonic on and is completely monotonic on . Using (2.3), it is easy to see that It is easy to see from this and the ring structure of completely monotonic functions on that is strictly completely monotonic on . Similarly, one shows that is completely monotonic on .
To prove the “only if” part of the assertions of the proposition, we note that from which we see easily by considering the case that fails to be increasing if . Similarly, by considering the case that fails to be decreasing if and this completes the proof.

4. Some Inequalities Involving Polygamma Functions

For integers and any real number , we define Here we set for convenience.

For , a result of Alzer and Wells [20, Theorem  2.1] asserts that the function is completely monotonic on if and only if and is completely monotonic on if and only if .

We denote and note that when . We now extend the result of Alzer and Wells to the following.

Theorem 4.1. Let be integers satisfying . The function is completely monotonic on . The function is also completely monotonic on when .

Proof. We first prove the assertion for with . The proof here uses the method in [20]. Using the integral representation (2.2) for and using for the Laplace convolution, we get where with It suffices to show that . By change of variables we can recast it as We now break the above integral into two integrals, one from 0 to and the other from to 1. We make a further change of variable for the first one and for the second one. We now combine them to get where the function is defined as in Lemma 2.7. Note that for , hence by Lemma 2.7, there is a unique number such that We further note it is shown in the proof of [20, Lemma  2.2] that the function is a decreasing function on so that for , Hence Note that the integral above is (by reversing the process above on changing variables) where the last step follows from the well-known beta function identity and the well-known fact for .
Now we prove the assertion for . In this case and we note that and we use this to write It follows that where the last inequality follows from Lemma 2.8.
It remains to show the assertion for . In this case we use the series representation in (2.2) for to get We note the following Binet-Cauchy identity: We now apply the above identity with to get We note that the second factor on the right-hand side above is completely monotonic on and also that Certainly each factor on the right-hand side above is completely monotonic on and it follows from the ring structure of completely monotonic functions on that the left-hand side expression in (4.21) is also completely monotonic on . Hence by the ring structure of completely monotonic functions on again we deduce that is completely monotonic on .