Further Inequalities Associated with the Classical Gamma Function

Kwara Nantomah

Turkish Journal of Analysis and Number Theory

Further Inequalities Associated with the Classical Gamma Function

Kwara Nantomah

Department of Mathematics, University for Development Studies, Navrongo Campus, Navrongo UE/R, Ghana

Abstract

In this paper, the authors present some double inequalities associated with certain ratios of the Gamma function. The results are further generalizations of several previous results. The approach is based on some monotonicity properties of some functions involving the generalized Gamma functions. At the end, some open problems are posed.

Cite this article:

  • Kwara Nantomah. Further Inequalities Associated with the Classical Gamma Function. Turkish Journal of Analysis and Number Theory. Vol. 3, No. 4, 2015, pp 111-115. http://pubs.sciepub.com/tjant/3/4/5
  • Nantomah, Kwara. "Further Inequalities Associated with the Classical Gamma Function." Turkish Journal of Analysis and Number Theory 3.4 (2015): 111-115.
  • Nantomah, K. (2015). Further Inequalities Associated with the Classical Gamma Function. Turkish Journal of Analysis and Number Theory, 3(4), 111-115.
  • Nantomah, Kwara. "Further Inequalities Associated with the Classical Gamma Function." Turkish Journal of Analysis and Number Theory 3, no. 4 (2015): 111-115.

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1. Introduction

Inequalities involving the classical Euler’s Gamma function has gained the attention of researchers all over the world. Recent advances in this area include those inequalities involving ratios of the Gamma function. In [1, 5, 6, 10] and [11-17][11], the authors established some interesting inequalities concerning such ratios, as well as some generalizations. By utilizing similar techniques, this paper seeks to present some new results generalizing the results of [11-17][11]. At the end, we pose some open problems involving the generalized Psi functions. In the sequel, we recall some basic definitions concerning the Gamma function and its generalizations. These definitions are required in order to establish our results.

The well-known classical Gamma function, and the classical Psi or Digamma function are usually defined for as:

The p-Gamma function, and the p-Psi function are defined for and as:

where and as . For more information on this function, see [9] and the references therein.

Also, the q-Gamma function, and the q-Psi function are defined for and as:

where and as .

See also [4, 5] and the references therein.

Similarly, the k-Gamma function, and the k-Psi function are defined for and as (see [2, 7]):

where and as .

Also, the (q,k)-Gamma function and the (q,k)-Psi function are defined for , and as [3]:

where is the k-generalized Pochhammer symbol and , as , .

Furthermore, the (p,q)-Gamma function and the (p,q)-Psi function are defined for , and as [8]:

and

where , and , as , .

As defined above, the generalized Psi functions: , , , and possess the following series forms (see [16, 17] and the references therein):

(1)
(2)
(3)
(4)
(5)

with denoting the Euler-Mascheroni’s constant.

2. Results

We now present our results. Let us begin with the following Lemmas pertaining to the results.

Lemma 2.1. Assume that , , and . Then,

Proof. By using equations (2) and (4) we obtain,

concluding the proof.

Lemma 2.2. Assume that , , and . Then,

Proof. By using equations (2) and (5) we obtain,

concluding the proof.

Lemma 2.3. Assume that , , , , and . Then,

Proof. By using equations (3) and (4) we obtain,

concluding the proof.

Lemma 2.4. Assume that , , , and . Then,

Proof. By using equations (3) and (5) we obtain,

concluding the proof.

Theorem 2.5. Let be a positive, increasing and differentiable function, and . Then for positive real numbers and such that , the inequalities:

(6)

hold true for .

Proof. Define a function for and by

Let . Then,

Then,

as a consequence of Lemma 2.1. That implies is non-increasing on . Hence is non-increasing and for we have,

establishing the inequalities in (6).

Theorem 2.6. Let be a positive, increasing and differentiable function, and . Then for positive real numbers and such that , the inequalities:

(7)

hold true for .

Proof. Define a function for and by

Let . Then,

Then,

as a consequence of Lemma 2.2. That implies is non-increasing on . Hence is non-increasing and for we have,

establishing the inequalities in (7).

Theorem 2.7. Let be a positive, increasing and differentiable function, , and . Then for positive real numbers and , the inequalities:

(8)

hold true for .

Proof. Define a function for , and by

Let . Then,

Then,

as a result of Lemma 2.3. That implies is increasing on . Hence is increasing and for we have,

establishing the inequalities in (8).

Theorem 2.8. Let be a positive, increasing and differentiable function, and . Then for positive real numbers and , the inequalities:

(9)

hold true for .

Proof. Define a function for and by

Let . Then,

Then,

as a result of Lemma 2.4. That implies is -increasing on . Hence is increasing and for we have,

establishing the inequalities in (9).

3. Concluding Remarks

In particular, if we let for and on the interval , then we recover the entire results of [17]. Also, by setting and on the interval , we obtain the results of [16]. The results [11][17] are therefore special cases of the results of this paper. For example, let for on the interval

. Then;

(i) by allowing in Theorem 2.5, we recover Theorem 3.7 of [13].

(ii) by allowing in Theorem 2.8, we recover Theorem 3.8 of [13].

(iii) by allowing in Theorem 2.6, we recover Theorem 3.9 of [13].

(iv) by allowing in Theorem 2.7, we recover Theorem 3.1 of [15].

This paper is a slightly modified version of preprint [18].

4. Open Problems

For , and , let , , and be the generalized Psi functions as defined in equations (1) – (5).

Problem 1: Under what conditions will the statements:

be valid?

Problem 2: Under what conditions will the statements:

be valid?

Competing Interests

The authors declare that there is no competing interest.

Acknowlegement

The authors are very grateful to the anonymous reviewers for their useful comments and suggestions which helped in improving the quality of this paper.

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