Some New Inequalities Using Nonintegral Notion of Variables

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


Introduction
It is important to note that the integral inequalities play a basic role in statistics, mathematics, sciences, and technology (SMST).As in [1][2][3][4][5][6][7][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][10][11][12].
As in [27][28][29][30][31][32], for a function gðvÞ ∈ L 1 ð½α, βÞ, the Hadamard fractional integral of order κ ≥ 0 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 βðl, mÞ, 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][34][35][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 κ ≥ 0 that generalizes Hadamard fractional integrals.

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 κ ≥ 0. Definition 1.For a function gðvÞ ∈ L 1 ð½α, βÞ, the generalized fractional integral of the Hadamard type with order κ ∈ ℝ + is given by where ΓðκÞ = Ð ∞ 0 e −u u κ−1 du represents the Gamma function as can be seen in [37,38] and many more.
Definition 4. For a r.v.Z having a positive p.d.f.g : ½α, β ⟶ ℝ + (α > 0), the fractional expectation function of order κ ≥ 0 is given as Definition 5.If EðZÞ symbolizes the expected value of the r.v.Z having a positive p.d.f.g : ½α, β ⟶ ℝ + with α > 0, then the fractional variance function having order κ of Z is given by where α < v < β.Definition 6.If EðZÞ symbolizes the expected value of the r.v.Z having a positive p.d.f.g : ½α, β ⟶ ℝ + with α > 0, then the fractional variance function having order κ of Z is given by Now by choosing different values of κ and λ, we have the following remarks.

Advances in Mathematical Physics
Proof.For the proof of the result, we begin by choosing the function H for x, y ∈ ðα, vÞ, α < v ≤ β as follows: where κ ≥ 0.
To prove (ii), we can write ð16Þ Now using ( 14) and ( 16), the part (ii) of the result follows.This completes the proof.
To prove part (ii), we use the truth that sup v,y∈½α,v jx − yj 2 = ðx − αÞ 2 and get Consequently, part (ii) of the result follows by employing (20) and (21).

Applications and Examples
Application 11.Consider the positive functions g and h on ½α, β such that for every u > α, where m ≥ 1; then, for every κ > 0, we have Solution 12. From (23), we see In a similar way, we see Consequently, multiplying these equations by ðln ðv/uÞÞ κ/λ−1 /uλΓðκÞ for u ∈ ðα, vÞ and then integrating the resulting identity over ðα, vÞ with respect to u 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).