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Some inequalities involving k-gamma and k-beta functions with applications
Journal of Inequalities and Applications volume 2014, Article number: 224 (2014)
In this paper, we present some inequalities involving k-gamma and k-beta functions via some classical inequalities like the Chebychev inequality for synchronous (asynchronous) mappings, and the Grüss and the Ostrowski inequality. Also, we give a new proof of the log-convexity of the k-gamma and k-beta functions by using the Hölder inequality.
In this section, we present some fundamental relations for k-gamma and k-beta functions introduced by the researchers [1–7]. The second and third section is devoted to the applications of some integral inequalities like the Chebychev, Grüss, and Ostrowski inequalities. In the last section, we prove the log-convexity of the k-gamma and k-beta functions.
Recently, Diaz and Pariguan  introduced the generalized k-gamma function as
and also gave the properties of said function. The is one parameter deformation of the classical gamma function such that as . The is based on the repeated appearance of the expression of the following form:
The function of the variable α given by the statement (2), denoted by , is called the Pochhammer k-symbol. We obtain the usual Pochhammer symbol by taking . The definition given in (1) is the generalization of and the integral form of is given by
From (3), we can easily show that
The same authors defined the k-beta function as
and the integral form of is
From the definition of given in (5) and (6), we can easily prove that
Note that when , .
2 Main results: inequalities via the Chebychev integral inequality
In this section, we prove some inequalities which involve k-gamma and k-beta functions by using some natural inequalities . The following result is known in the literature as the Chebychev integral inequality for synchronous (asynchronous) functions. Here, we use this result to prove some k-analog inequalities.
Lemma 2.1 Let be such that for all and h, , hf, and hg are integrable on I. If f, g are synchronous (asynchronous) on I, i.e.,
This lemma can be proved by using the Korkine identity ,
Theorem 2.2 If m, n, p, and q are positive real numbers satisfying the condition
then, for the k-beta function, we have the inequality
Proof For , consider the mappings given by
Now, differentiation of f and g gives
As , so using (22) and (23), we see that the mappings f and g are synchronous (asynchronous) having the same (opposite) monotonicity on and h is non-negative on . Thus, using the Chebychev integral inequality for the functions f, g, and h defined above, we have
Applying (6), we get the required inequality (24). □
Corollary 2.3 For positive real numbers m, n, p, and q, we have
Proof Using (5) and inequality (24), we have
Corollary 2.4 For , the following inequality holds for the k-beta function:
Proof Setting and in Theorem 2.2, we get
Thus, Corollary 2.4 follows. We have
Remarks 2.5 By (5) and (26), we can deduce the following inequality for the k-gamma function:
Setting and in (27), we get
From the above result, we conclude that for two positive numbers s and t, the geometric mean of and is greater than or equal to (arithmetic mean of s and t).
Now, the Chebychev inequality is used for an infinite interval. For this purpose, see the following theorem employing the inequality .
Theorem 2.6 Let m, p, and r be positive real numbers such that . If , then
Proof For , define the mappings given by
If , holds and , then we can assert that the mappings f and g are synchronous (asynchronous) . Thus, using the Chebychev inequality for the interval along with the functions f, g, and h defined above, we can write
By (3), we get the required Theorem 2.6. □
Corollary 2.7 If p, m, and r are positive real numbers satisfying the conditions of Theorem 2.6, then we deduce
Proof Using the property and from the inequality (28), we can derive Corollary 2.7. □
Corollary 2.8 If and with , then
Proof Setting and in Theorem 2.6, we get and inequality (28) provides the desired Corollary 2.8. □
Definition 2.9 Two positive real numbers a and b are said to be similarly (oppositely) unitary if (see )
Theorem 2.10 If are similarly (oppositely) unitary, then
Proof For , consider the mappings defined by
If the condition holds and , then clearly the mappings f and g are synchronous (asynchronous) on . Thus, by the Chebychev integral inequality along with the functions f, g, and h defined above, we have
By the definition of the k-gamma function and then using (8), we have
Corollary 2.11 If the condition holds and , then we have
Proof Obvious result from (5) and Theorem 2.10. □
Remarks 2.12 The results proved here are k-analog of theorems as given in . Using , we have the results about classical gamma and beta functions.
Theorem 2.13 If a, b, and k are positive real numbers such that a and b are similarly (oppositely) unitary, then
Proof For , consider the mappings defined by
If the conditions of Theorem 2.10 hold and , then clearly the mappings f and g are synchronous (asynchronous) on . Thus, by the Chebychev integral inequality along with the choice of the functions f, g, and h defined above, we have
By the definition of the k-gamma function and then using (8) and (10), we have
Corollary 2.14 For all and , prove that
Proof Replacing b by in the inequality (33), we get
By multiplying the above inequalities, we have the desired Corollary 2.14. □
3 Main results via Grüss and Ostrowski inequalities
In 1935, Grüss established an integral inequality which provides an estimation for the integral of a product in terms of the product of integrals . Here, we use this inequality to prove some inequalities involving k-gamma and k-beta functions. We also use the weighted version of the said inequality which allows us to obtain the inequalities directly for k-gamma function. The following lemma is used to prove some k-analog inequalities.
Lemma 3.1 Let f and g be two functions defined and integrable on . If m, M, n, and N are given real constants such that and for all , then
and the constant is the best possible (see ).
Theorem 3.2 Let p, q, r, s, and k be positive real numbers, then
Proof Consider the functions defined by
For the application of the Grüss inequality, we have to find the minima and maxima of (). Thus
Here, we see that the solution of in the interval is . Also, on and on . We conclude that is the maximum point in the interval and consequently
Hence, by the Grüss inequality, we get
Rearranging the terms on left-hand side and using (6) with simple algebraic computation, we reach the required proof. □
Theorem 3.3 Let p, q, and k be positive real numbers, then prove that
and an equivalent statement is given by
Proof Consider the functions defined by
For minima and maxima of and , we have
By using the Grüss inequality, we get
Using the definition of the k-beta function, we have
Algebraic computation provides the equivalent inequality (37). □
Corollary 3.4 If we use (5), the inequality (36) yields
By using (8), we get the following inequality:
Now, we discuss the weighted version of the Grüss inequality which is used to generalize the previous Theorems 3.2 and 3.3. The weighted version is given in the following lemma and generalized results in the form of propositions.
Lemma 3.5 Let f and g be two functions satisfying the conditions of Theorem 3.2 and is such that , then
and the constant is best possible.
Proof Similar to the classical one (see ). □
Proposition 3.6 Let m, n, p, q, and k be positive real numbers and ; then we have
Proof Straightforward by considering the choice of the following functions along with Lemma 3.5 which generalizes Theorem 3.2:
Proposition 3.7 Let p, q, and k be positive real numbers and , then
Proof Using Lemma 3.5 by considering the choice of functions defined by
we can prove Proposition 3.7, which is the generalization of Theorem 3.3. □
The weighted version of the Grüss inequality allows us to obtain the inequalities directly for k-gamma function (see the following theorem).
Theorem 3.8 Let a, b, and c are positive real numbers, then prove that
Proof Let the mapping be defined on , then
gives the unique solution , which implies that is an increasing function on and decreasing on . Thus has a maximum value at i.e., . Using Lemma 3.5, we get
which (for all ) is equivalent to
Since all the integrals involved here are convergent on , we have
Now, a simple change of variable , above integrals can be changed into k-gamma functions as
Similarly, we have
From (42) along with the above results, we get Theorem 3.8. □
Now, we mention here another inequality which is known in literature as the Ostrowski inequality. The following lemmas contain the said integral inequality  which is used to prove the inequalities involving k-beta function.
Lemma 3.9 Let be continuous on and differentiable in , whose derivative is bounded on and let . Then, for all , we have
The constant is sharp in the sense that it cannot be replaced by a smaller one and the following lemma is the generalization of the inequality (43) which has been proved in .
Lemma 3.10 Let be a L-Lipschitzian mapping on , i.e.,
Then, for all , we have
The constant is best possible.
Remarks 3.11 If we assume that is differentiable on and the derivative is bounded on we can put instead of L the infinity norm obtaining the estimation due to Dragomir-Wang in . Now we are able to prove our result for the k-beta function.
Theorem 3.12 If and and , then we have the following inequality:
Proof Consider the mapping , defined by , . For , differentiation gives
The solution of in the interval is . Also, on and on . This shows that is the maximum point, so
Consequently, for all , we have
Thus, for , we get
Taking the function , and using the inequalities (47) and (45) along with Remarks 3.11, we have
Corollary 3.13 The best inequality that can be obtained for and is
Proof Setting , we get the desired result. □
Here, we use the Ostrowski inequality for the mappings of bounded variation  involving k-beta function. For this purpose, we need the following lemmas.
Lemma 3.14 Let be a mapping of bounded variation on . Then, for all , we have the inequality (see )
where indicates the total variation of u and the constant is the best possible.
Lemma 3.15 If is continuous and differentiable on , is continuous on and , then, for all , we have the inequality (see )
Theorem 3.16 If and , , then we have the following inequality:
Proof Consider the mapping as in Theorem 3.12. Now, we have
Also, for all , so
Now, applying the previous lemmas for , , we conclude
By definition of the k-beta function, we have the required Theorem 3.16. □
Corollary 3.17 Let , then the best inequality that can be obtained from inequality (50) is
Proof Setting in the inequality (50), we get the desired result. □
4 Log-convexity of the k-gamma and k-beta functions
Many authors, see [15–18] and the references therein, have worked on convexity, log-convexity, and exponential convexity of different functions including the Euler gamma function. Lately, Diaz and Pariguan  worked on the convexity of k-gamma function and proved that the said function is logarithmically convex. They used the limit form of k-gamma function for this purpose. Here, we give a new technique to prove the log-convexity of the k-gamma function. Also, we prove the log-convexity of the k-beta function which is the k-analog result .
Definition 4.1 In , a function is said to be convex if for any and
The above definition shows that when we move from x to y, the line joining the points and lies always above the graph of f.
Definition 4.2 If and logf is convex, then f is called a log-convex function i.e., (an interval) and , we have
Lemma 4.3 (Hölder inequality)
If p and q are positive real numbers satisfying the condition , then for integrable functions , we have
Proof The proof of the above inequality is available in . □
Theorem 4.4 For , prove that is log-convex or is convex (proved in , but we give a new proof here).
Proof Let p and q be positive real numbers satisfying the condition . By the definition of , we have
By Lemma 4.3, we conclude that
Let , , then and
for , is convex i.e., is log-convex. □
Remarks 4.5 By Theorem 4.4, the function is log-convex. Also, every log-convex function is convex , so the k-gamma function is convex.
Theorem 4.6 For , prove that the function is logarithmically convex on as a function of two variables.
Proof Let and with , we have
Using (6) on right-hand side of the inequality (55), we get
Thus, we get
Here, , , then , which shows the logarithmic convexity of on . □
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The authors are grateful to the anonymous referees for their helpful comments and suggestions to improve the article.
The authors declare that they have no competing interests.
The main idea of this paper was proposed by AR and SM. The both authors contributed equally to the writing of this paper. The authors NS and FS read and approved the final manuscript.
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Rehman, A., Mubeen, S., Sadiq, N. et al. Some inequalities involving k-gamma and k-beta functions with applications. J Inequal Appl 2014, 224 (2014). https://doi.org/10.1186/1029-242X-2014-224