1. Introduction

We begin by outlining the following basic definitions well-known in literature.

The celebrated classical Euler’s Gamma function, is defined for as

The q-Gamma function, is defined for and as (see ^[2])

Also, the k-Gamma function, was defined by Diaz and Pariguan ^[1] for and as

Diaz and Teruel ^[5] further defined the (q,k)-Gamma function for , and as

where

is the k-generalized Pochhammer symbol.

Furthermore, Krasniqi and Merovci ^[4] defined the (p,q)-Gamma function for , and as

where

The psi function, otherwise known as the digamma function is defined as the logarithmic derivative of the Gamma function. That is,

The q-digamma function, k-digamma function, (p,q)-diagamma function and (q,k)-digamma function are similarly defined as follows:

and

It is common knowledge that these functions exhibit the following series charaterizations (see also [7-12]^[7]):

(1)

(2)

(3)

(4)

where represents the Euler-Mascheroni’s constant.

Of late, the following double inequalities were presented in ^[7] by the use of some monotonicity properties of some functions related with the Gamma function.

(5)

for , , and .

(6)

for , , and .

(7)

for ,, , and .

(8)

for ,, and .

Results of this form can also be found in ^{[8, 9, 10, 11, 12]}. By utilizing similar techniques as in the previous results, this paper seeks to provide some generalizations of the above inequalities. We present our results in the following sections.

2. Supporting Results

We begin with the following Lemmas.

Lemma 2.1. Suppose that ,, ,, and . Then,

Proof. From the characterization in equations (1) and (3) we obtain,

We conclude the proof by substituting by .

Lemma 2.2. Suppose that ,, ,, and . Then,

Proof. From the characterization in equations (1) and (4) we obtain,

We conclude the proof by substituting by .

Lemma 2.3. Suppose that ,, , ,, and . Then,

Proof. From the characterization in equations (2) and (3) we obtain,

We conclude the proof by substituting by .

Lemma 2.4. Suppose that ,, , , and . Then,

Proof. From the characterization in equations (2) and (4) we obtain,

We conclude the proof by substituting by .

3. Main Results

We now present our results in the following Theorems.

Theorem 3.1. Define a function for and by

(9)

where , , , are positive real numbers such that . Then, is non-increasing on and the inequalities:

(10)

are valid for each .

Proof. Let for every . Then

Then,

as a result of Lemma 2.1. That implies is non-increasing on . Consequently, is non-increasing on and for each we have,

yielding equation (10).

Theorem 3.2. Define a function for and by

(11)

where , , , are positive real numbers such that . Then, is non-increasing on and the inequalities:

(12)

are valid for each .

Proof. Let for every . Then

Then,

as a result of Lemma 2.2. That implies is non-increasing on . Consequently, is non-increasing on and for each we have,

yielding equation (12).

Theorem 3.3. Define a function for , , and by

(13)

where , , , are positive real numbers. Then, is increasing on and the inequalities:

(14)

are valid for each .

Proof. Let for every . Then

Then,

as a result of Lemma 2.3. That implies is non-increasing on . Consequently, is non-increasing on and for each we have,

yielding equation (14).

Theorem 3.4. Define a function for , and by

(15)

where , , , are positive real numbers. Then, is increasing on and the inequalities:

(16)

are valid for each .

Proof. Let for every . Then

Then,

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

yielding equation (16).

4. Conclusion

If we fix in inequalities (10), (12), (14) and (16), then we respectively obtain the inequalities (5), (6), (7) and (8) as special cases. By this, the previous results ^[7] have been generalized.

Competing Interests

The authors have no competing interests.

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