Certain Hadamard Proportional Fractional Integral Inequalities

: In this present paper we study the non-local Hadmard proportional integrals recently proposed by Rahman et al. (Advances in Difference Equations, (2019) 2019:454) which containing exponential functions in their kernels. Then we establish certain new weighted fractional integral inequalities involving a family of n ( n ∈ N ) positive functions by utilizing Hadamard proportional fractional integral operator. The inequalities presented in this paper are more general than the inequalities existing in the literature.


Introduction
The field of fractional integral inequalities play an important role in the field of differential equations and applied mathematics. These inequalities have many applications in applied sciences such as probability, statistical problems, numerical quadrature and transform theory. In the last few decades, many mathematicians have paid their valuable considerations to this field and obtained a bulk of various fractional integral inequalities and their applications. The interested readers are referred to the work of [1][2][3][4][5] and the references cited therein. A variety of different kinds of certain classical integral inequalities and their extensions have been investigated by considering the classical Riemann-Liouville (RL) fractional integrals, fractional derivatives and their various extensions. In [6], the authors presented integral inequalities via generalized (k, s)-fractional integrals. Dahmani and Tabharit [7] proposed weighted Grüss-type inequalities by utilizing Riemann-Liouville fractional integrals. Dahmani [8] presented several new inequalities in fractional integrals. Nisar et al. [9] investigated several inequalities for extended gamma function and confluent hypergeometric k-function. Gronwall type inequalities associated with Riemann-Liouville k and Hadamard k-fractional derivative with applications can be found in the work of Nisar et al. [10]. Rahman et al. [11] presented certain inequalities for generalized fractional integrals. Sarikaya and Budak [12] investigated Ostrowski type inequalities by employing local fractional integrals. The generalized (k, s)-fractional integrals and their applications can be found in the work of Sarikaya et al. [13]. In [14], Set et al. proposed the generalized Grüss-type inequalities by employing generalized k-fractional integrals. Set et al. [15] have introduced the generalized version of Hermite-Hadamard type inequalities via fractional where δ ≥ ξ > 0, τ, ν, σ > 0.

Theorem 3.
Let the two functions U and V be positive and continuous on the interval [1, ∞) such that the function U is decreasing and the function V is increasing on [1, ∞) and assume that the function W : [1, ∞) → R + is positive and continuous. Then for all θ > 1, the following inequality for Hadamard proportional fractional integral operator (7) holds; where δ ≥ ξ > 0, τ, ν, σ > 0.
Proof. Since the two functions U and V are positive and continuous on (16)) and integrating the resultant estimates with respect to ρ over (1, θ), we get which in view of (7) becomes, Again, multiplying (22) by F (θ, ζ) (where F (θ, ζ) can be obtained from (16)) and integrating the resultant estimates with respect to ζ over (1, θ) and then employing (7), we get which gives the desired assertion (20).

Theorem 4.
Let the two functions U and V be positive and continuous on the interval [1, ∞) such that the function U is decreasing and the function V is increasing on [1, ∞) and assume that the function W : [1, ∞) → R + is positive and continuous. Then for all θ > 1, the following inequality for Hadamard proportional fractional integral operator (7) holds; Proof. Taking product on both sides of (22) by G(θ, ζ) = 1 where λ > 0, ξ > 0 and ζ ∈ (1, θ), θ > 1. and integrating the resultant estimates with respect to ζ over (1, θ), we have Consequently, in view of (7) it can be written as, which gives the desired inequality (23).
Next, we present some fractional proportional inequalities for a family of n positive functions defined on [1, ∞) by utilizing Hadamard proportional fractional integral (7).

Theorem 7.
Let the functions U j , (j = 1, 2, · · · , n) and V be positive and continuous on the interval [1, ∞) such that the function V is increasing and the function U j for j = 1, 2, · · · , n are decreasing on [1, ∞).