Keywords: Gamma function, qGamma function, kGamma function, (p,q)Gamma function, (q,k)Gamma function, inequality
Turkish Journal of Analysis and Number Theory, 2014 2 (6),
pp 226229.
DOI: 10.12691/tjant267
Received October 10, 2014; Revised November 26, 2014; Accepted December 14, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
We begin by outlining the following basic definitions wellknown in literature.
The celebrated classical Euler’s Gamma function, is defined for as
The qGamma function, is defined for and as (see ^{[2]})
Also, the kGamma 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 kgeneralized 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 qdigamma function, kdigamma 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 [712]^{[7]}):
 (1) 
 (2) 
 (3) 
 (4) 
where represents the EulerMascheroni’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 nonincreasing 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 nonincreasing on . Consequently, is nonincreasing 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 nonincreasing 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 nonincreasing on . Consequently, is nonincreasing 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 nonincreasing on . Consequently, is nonincreasing 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 nonincreasing on . Consequently, is nonincreasing 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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