Abstract
The object of this paper is to present an extension of the classical Hadamard fractional integral. We will establish some new results of generalized fractional inequalities.
1. Introduction
It is important to note that the integral inequalities play a basic role in statistics, mathematics, sciences, and technology (SMST). As in [1–8], it has proven to be of great importance from the past few decades. The formation of fractional calculus has straight impact on the theory utilizing the solution of various spaces in SMST and to prove its efficacy, various statements and applications of fractional derivatives have been constructed. Authors Riemann–Liouville and Grunwald–Letnikov are well known in this filed. Caputo reformulated the classical statement of the Riemann–Liouville fractional derivative for finding solutions of fractional differential equations using initial conditions. The notion of fractional calculus given by Leibniz was studied by Grunwald–Letnikov in a different structure [9–12].
Recently, in [13–17] and [18], the development between probability theory and fractional calculus was given, and the results of the classical approach were extended. Also, analysis and observations on this direction and several purposes have been found in the concrete problems which include applied mathematics and fluid mechanics as in [2, 18–26], and many others.
As in [27–32], for a function , the Hadamard fractional integral of order is given as follows: which differs from Riemann-Liouville and Caputo’s definition in the sense that the kernel of integral (1) contains a logarithmic function of an arbitrary exponent.
We need the following definition while determining some application, and it is called the Beta function.
As in [22], the Beta function, symbolized by , is given as
The basic notion of generalization of special functions using a kind of new parameter fascinated many researchers and mathematicians. More details about fractional integrals can be found in [33–36] and others as cited in the text. Accordingly, our main scenario in this paper is to extend the idea of a new fractional integration with parameter that generalizes Hadamard fractional integrals.
2. Main Results
In this section, we shall be dealing with the new generalized type of results to random variables of a continuous type of fractional integral order .
Definition 1. For a function , the generalized fractional integral of the Hadamard type with order is given by where represents the Gamma function as can be seen in [37, 38] and many more.
Definition 2. For a r.v. having positive p.d.f. (), we define the fractional expectation function of order as where .
Definition 3. For a r.v. having a positive p.d.f. (), the fractional expectation function of of order is given as where .
Definition 4. For a r.v. having a positive p.d.f. (), the fractional expectation function of order is given as
Definition 5. If symbolizes the expected value of the r.v. having a positive p.d.f. with , then the fractional variance function having order of is given by where .
Definition 6. If symbolizes the expected value of the r.v. having a positive p.d.f. with , then the fractional variance function having order of is given by Now by choosing different values of and , we have the following remarks.
Remark 7. (R1) Choosing and , the classical expectation of r.v. will be deduced.
(R2) Choosing and , the classical variance of r.v. will be deduced.
(R3) Choosing , we reach to the definition of [31].
Theorem 8. Let the continuous r.v. be with p.d.f. . Then, for all .
Proof. By definition, we have
Theorem 9. Define a r.v. having a p.d.f. . Then, we have the following inequalities: (i)holds if and for all , , and . (ii)holds.
Proof. For the proof of the result, we begin by choosing the function for as follows: where .
Now on both sides of (11), we multiply by , where the is a function , and then integrating the resulting identity from to , we see
Now multiplying (12) by for , and then integrating the resulting identity over with respect to , we see
Putting and in (13), we see
But we can also write that as
Consequently, part (i) follows from (14) and (15).
To prove (ii), we can write
Now using (14) and (16), the part (ii) of the result follows. This completes the proof.
Theorem 10. Let the continuous r.v. be with p.d.f. . Then (i)the inequalityholds for , and and for all and (ii)the inequalityholds for and .
Proof. To prove (i), we multiply (12) by for both sides and get
Putting , , and in (19), we see
But we can also write
Consequently, part (i) of the result follows from (20) and (21).
To prove part (ii), we use the truth that and get
Consequently, part (ii) of the result follows by employing (20) and (21).
3. Applications and Examples
Application 11. Consider the positive functions and on such that for every , with where ; then, for every , we have
Solution 12. From (23), we see
In a similar way, we see
Consequently, multiplying these equations by for and then integrating the resulting identity over with respect to yield
Now on multiplying (27) and (28), we see
Consequently, the result follows by using Minkowski’s integral inequality on the right-hand side of (29).
Example 1. Consider the function , we see Choosing , for with , we see with the help of (2) that
Data Availability
No data were used in this study.
Conflicts of Interest
The authors declare that there is no conflict of interest.