Some inequalities involving k-gamma and k-beta functions with applications - II
© Rehman and Mubeen; licensee Springer. 2014
Received: 13 August 2014
Accepted: 23 October 2014
Published: 4 November 2014
In this paper, we present some inequalities involving k-gamma and k-beta functions via some classical inequalities, like Chebyshev’s inequality for synchronous (asynchronous) mappings, Grüss’, and Ostrowski’s inequality. Also, we give applications of k-beta function in probability distributions. Most of the inequalities produced in this paper are the k-analogs of existing results. If , we have the classical one.
In this section, we present some fundamental relations for k-gamma and k-beta functions introduced in [1–7]. In Section 2, we introduce some k-analog properties of the mapping , which is helpful in coming sections. Sections 3 to 5 are devoted to the applications of some integral inequalities like Chebyshev’s, Grüss’, and Ostrowski’s inequality for k-beta mappings. In the last section, we give the applications of the said function for the probability distribution and the probability density function.
Note that when , .
For more details about the theory of k-special functions like the k-gamma function, the k-polygamma function, the k-beta function, the k-hypergeometric functions, solutions of k-hypergeometric differential equations, contiguous functions relations, inequalities with applications and integral representations with applications involving k-gamma and k-beta functions, k-gamma and k-beta probability distributions, and so forth (see [8–15]).
2 Main results: some k-analog properties of the mapping
Remark If , we have the properties of the mapping given in .
3 Chebyshev type inequalities involving k-beta and k-gamma functions
In this section, we prove some inequalities which involve k-gamma and k-beta functions by using some natural inequalities . The following result is well known in the literature as Chebyshev’s integral inequality for synchronous (asynchronous) functions. Here, we use this result to prove some k-analog inequalities.
provided that and , are differentiable and the first derivatives are bounded on I.
for all , we can deduce the desired inequality (38). □
Proof Just use in Theorem 3.2 to get the required corollary. □
which will be equivalent to Theorem 3.4 by applying (6) on both sides of the above inequality. □
By (8) and (10), inequality (41) can be obtained and some algebraic calculations give the desired inequality (42). □
4 Some other inequalities for k-beta mappings
In 1935, Grüss established an integral inequality which gives an estimation for the integral of a product in terms of the product of integrals . We use the following lemma  to prove our next theorem which is based on the Grüss integral inequality.
Now, by Lemma 4.1, we get the required result. □
The following inequality of Grüss type has been established in .
where , .
we have our required result. □
Remarks If we use , inequalities (43) and (44) are the results for the classical beta function proved in .
By (6) and the fact , we have the inequality (45). Also, and . Thus, using Lemma 4.6 we have the inequality (46). □
5 Main results: via Ostrowski’s inequality
In this section, we use the integral inequality which is known in the literature as Ostrowki’s inequality . The following lemma concerning Ostrowski’s inequality for absolutely continuous mappings whose derivatives belong to spaces hold [24, 25]. Here, we give some lemmas which are helpful for the results involving k-beta mapping.
For the application of the above inequalities to some numerical quadrature rules, we have the following lemma.
where (). Lemmas 5.1 and 5.2 are proved in and the best quadrature formula that can be obtained from the above result is one for which , , and is given in the following corollary.
We are now able to apply the above results for Euler’s k-beta mapping.
provided that .
Using (48) and (49), we get the desired Theorem 5.4. □
Now, we have the result concerning the approximation of the k-beta function in terms of the Riemann sums.
where () and .
Proof Taking , , along with Lemma 5.2 we get Theorem 5.5. The proof of Lemma 5.2 is available in , so details are omitted. □
6 Inequalities in probability theory and applications for k-beta function
Here, we give some applications of the Ostrowski type inequality for the k-beta function and cumulative distribution functions. For this purpose, we need some basic concepts of random variable, distribution function, probability density function and expected values.
A process which generates raw data is called an experiment and an experiment which gives different results under similar conditions, even though it is repeated a large number of times, is termed a random experiment. A variable whose values are determined by the outcomes of a random experiment is called a random variable or simply a variate. The random variables are usually denoted by capital letters, X, Y, and Z, while the values associated to them by corresponding small letters x, y, and z. The random variables are classified into two classes namely discrete and continuous random variables.
A random variable that can assume only a finite or countably infinite number of values is known as a discrete random variable, while a variable which can assume each and every value within some interval is called a continuous random variable. The distribution function of a random variable X, denoted by , is defined by i.e., the distribution function gives the probability of the event that X takes a value less than or equal to a specified value x.
which shows the area under the curve between and .
Moments about any value is the r th power of the deviation of variable from A and is called the r th moment of the distribution about A.
Moments about is the r th power of the deviation of variable from 0 and is called the r th moment of the distribution about 0.
Moments about mean i.e., is the r th power of the deviation of variable from mean and is called the r th moment of the distribution about mean. If a random variable X assumes all the values from a to b, then for a continuous distribution, the r th moments about the arbitrary number A and 0, respectively, are given by and (see [26–28]).
for all . In particular, for the intermediate point of the interval , i.e., at , we have the remaining results of Theorem 6.5. □
for all , where .
Proof Using Lemma 6.6 along with the k-beta random variable and defined in (50), (51), and (53) for the expected values, we get the required Theorem 6.7. □
The authors are grateful to the editor for suggestions, which improved the contents of the article.
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