- Open Access
Some inequalities involving k-gamma and k-beta functions with applications
© Rehman et al.; licensee Springer. 2014
- Received: 26 February 2014
- Accepted: 22 May 2014
- Published: 3 June 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.
Note that when , .
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.
Applying (6), we get the required inequality (24). □
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 .
By (3), we get the required Theorem 2.6. □
Proof Using the property and from the inequality (28), we can derive Corollary 2.7. □
Proof Setting and in Theorem 2.6, we get and inequality (28) provides the desired Corollary 2.8. □
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.
By multiplying the above inequalities, we have the desired Corollary 2.14. □
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.
and the constant is the best possible (see ).
Rearranging the terms on left-hand side and using (6) with simple algebraic computation, we reach the required proof. □
Algebraic computation provides the equivalent inequality (37). □
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.
and the constant is best possible.
Proof Similar to the classical one (see ). □
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).
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.
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 .
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.
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.
where indicates the total variation of u and the constant is the best possible.
By definition of the k-beta function, we have the required Theorem 3.16. □
Proof Setting in the inequality (50), we get the desired result. □
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 .
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.
Lemma 4.3 (Hölder inequality)
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).
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.
Here, , , then , which shows the logarithmic convexity of on . □
The authors are grateful to the anonymous referees for their helpful comments and suggestions to improve the article.
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