The fractional structures of variables using Riemann–Liouville notion have been analyzed by various authors. The novel idea of this article is to introduce the new notion of weighted behavior on random variables using integral inequalities. In view of these, we obtain some new generalized fractional inequalities by using this new fractional integration of continuous random variables.
In recent times, the integral inequalities play a vital role in science, engineering, and technology. The authors are much attracted to study and analyze the structure in the fields like physics, statistics, biology, chemistry, and engineering as witnessed in refs. [1,2,3] and many others. The basic concern of fractional calculus has a straight knock on the solution of various problems of fast growing sciences to stimulate much interest in such field and to show its visibility in them. The different types of procedures and applications of fractional derivatives have been developed as can be found in refs. [4,5, 6,7,8, 9,10]. In this direction, Riemann–Liouville (RL) and Grunwald–Letnikov are most notable authors. It was Leonhard Euler in around 1720s gave the concept of how to extend the factorial to noninteger values. This developed a rich theory utilizing over the scientific world.
For a real number , the Euler gamma function for can be expressed as an improper integral:
It is observed that
It was Grunwald–Letnikow who forwarded the definition of fractional derivative order as follows:
For , we see from (2) that
Using the notion of theory of probability concept, we have following:
The classical approach of the Riemann–Liouville fractional derivative was reformulated by Caputo and he then gave the solution of fractional differential equations under given initial conditions. It was Grunwald–Letnikov who later put forward the fractional calculus given by Leibnitz in a new way. Importance of fractional calculus by using its execution to real-world issues in applied analyses, statistics, physics, and fluid mechanics eta cetera can be witnessed in refs. [18,19,20, 21,22,23, 24,25,26, 27,28,29] and many others and the classical approach was generalized.
The left and right Riemann–Liouville fractional integral of order , respectively, are given by
where is called as the Gamma function and .
For and , we have following well-known results:
For , the space of real-valued Lebesgue measurable functions on such that
For and , we consider the space of all real-valued Lebesgue measurable functions on such that
and choosing , we define it as follows:
By choosing with , then gets coincide to -space, and when we assume for then space reduced to the classical space
Let and , the generalized Riemann–Liouville fractional integrals and with order are given by
Throughout the article, random variable of will be abbreviated by r.v. and as a positive continuous function.
Let be a r.v. having a positive p.d.f. given on . Then, for and , the -weighted fractional expectation function of order is defined as follows:
We define the -weighted fractional expectation for a r.v. of order as follows:
Note by choosing , the aforementioned definitions reduces to following:
Let be a r.v. with a positive p.d.f. defined on . Then, for , we define the -weighted fractional expectation function of order as follows:
For mathematical expectation of r.v. having a positive p.d.f. . Then, on , and , the -weighted fractional variance function of order is defined as follows:
If , then -weighted fractional variance function of order is given as follows:
We have following important points:
2 Main results
This section will be concerning the new generalization of outcomes of continuous r.v. having fractional integral order.
Let be a r.v. with p.d.f. Then for each , and , we see
(i) the inequality
holds provided and
(ii) the inequality
Define a function for as follows:
Now multiplying (22) by both sides, where the is a function with , and then integrating the resulting identity from to and have
Now multiplying (23) by for , and then integrating the resulting identity over with respect to , we see
Now in (24), choosing and , , we see
But, on the other hand, we see
(ii) We have
If for every and , the continuous r.v. having p.d.f. in , then,
(i) the inequality
gets satisfied if and
(ii) the inequality
gets satisfied for any .
For a r.v. having a p.d.f. Then
(i) for all , , and , we have
holds for and
(ii) the inequality
holds for any , , and .
From (22), we can write
For , put , in (32), we see
But we also see
Consequently, using this and by involving (33) will establish part (i) of the result.
Now to establish (ii), we shall make use of following
Let be a r.v. having p.d.f. Then
for every , and .
From Theorem 3.1 of ref. , one can write
For , we choose and , then and . Thus, from (36), we can have
This yields that
This completes the proof.□
Now choosing and , we have following corollary:
For a p.d.f. g of r.v. , we have
for and .
For a r.v. with p.d.f. . Then
for every , , and .
From Theorem 3.4 of ref. , we can write
Choosing and in (40) yields
This yields us
Consequently, the result is established by simple calculation with utilizing the values of and from Theorem 2.6.□
In this paper, we have presented various concepts of fractional calculus. Also, some new generalizations of outcomes for continuous random variables having fractional order have been given. Furthermore, certain definitions like ℧-weighted fractional expectation and variance, have been introduced and their various properties have been studied. Moreover, new bounds and inequalities have been established. The consequences of the results obtained in this manuscript are more general and extensive than the pre-existing known results.
The authors are thankful to the anonymous reviewers for their valuable comments and suggestions towards the improvement of the article.
Funding information: The authors state no funding involved.
Author contributions: The author have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
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