Certain generalized fractional integral inequalities

1 Department of Mathematics, College of Arts and Science, Prince Sattam bin Abdulaziz University, Wadi Al dawaser, Riyadh region 11991, Saudi Arabia 2 Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal, Upper Dir, Khyber Pakhtoonkhwa, Pakistan 3 College of Computer and Information Sciences, Majmaah University, Al-Majmaah 11952, Saudi Arabia 4 Department of Mathematics, Faculty of Science, Kafrelshiekh University, Kafrelshiekh, Egypt


Introduction
Fractional integral inequalities (FII in short) have made a great impact on scientists and mathematicians because of its potential applications in various fields. This subject plays a vital role in the development of differential equations and related problems in applied mathematics. In recent few decades, a variety of various integral inequalities and their generalizations have been established by utilizing fractional integral, fractional derivative operators and their generalizations are found in [4-6, 10, 14-16, 19-21, 29, 35]. Also, the applications of (k, s)-Riemann-Liouville (R-L) fractional integral is found in [30]. In the past few years, various researchers have established the generalization of some classical inequalities by using different mathematical techniques. The generalized Hermite-Hadamard type inequalities with fractional integral operators and Hermite-Hadamard type inequalities by using the generalized k-fractional integrals are given in [34] and [2] respectively.
In [1], the authors established FII for a class of n decreasing positive functions where n ∈ N by using (k, s)-fractional integral operator. Recently, the researchers [17,18,[22][23][24][25][26] have established certain inequalities by employing some recent type(proportional and conformable) of fractional integrals. Without any doubt one can state that fractional and k-fractional calculus have become a very powerful tool for the modern studies, see for example [36,37]. To move towards our main results, we recall the following definitions [9,27,31]. Definition 1.1. Let f (τ), τ ≥ 0, real valued function, is said to be in the space Definition 1.2. Let ν,ν, ξ,ξ ∈ C such that R(ϑ) > 0 and x ∈ R. Then MSM fractional integral is defined by where F 3 (.) represents the Appell function (or Horn function) which is given in [8] as x m y n m!n! , max{|x|, |y|} < 1, The operator (1.1) is introduced in [13] and extended in [31,32]. The use of this function in connection with special functions is appeared in many recent papers [3,11,12].

Main results
In this section, we employ the MSM fractional integral operator to establish the generalization of some classical inequalities. Recalling the following Theorem which will be used to establish our main result.
Theorem 1. (see [28], Theorem 1) If ν, ν , ξ, ξ , η ∈ R such that η > max{ν, ν , ξ, ξ } > 0, then the following inequality holds Theorem 2. Let g be a positive continuous and decreasing function on the interval [a, b]. Let ν, ν , ξ, ξ , η ∈ R such that η > max{ν, ν , ξ, ξ } > 0, a < x ≤ b, ϑ > 0 and σ ≥ γ > 0 . Then for MSM fractional integral operator (1.1), we have Proof. Since g be a positive continuous and decreasing functions on the interval [a, b]. Therefore, we have (2.5) In view of Theorem 1, we observe that the function F(x, t) remain positive for all t ∈ (a, x), x > a, since each term of the above function is positive in view of conditions stated in Theorem 2. Therefore multiplying (2.4) by Integrating (2.6) with respect to t over (a, x), we have where F(x, ρ) is defined by (2.5) and integrating the resultant identity with respect to ρ over (a, x), we get It follows that

Dividing the above equation by
Remark 2.1. The inequality in Theorem 2 will reverse if g is an increasing function on the interval [a, b].
Proof. By multiplying both sides of (2.8) by where F(x, ρ) is defined by (2.5) and integrating the resultant identity with respect to ρ over (a, x), we have Hence, dividing (2.10) by we get the required results.

Theorem 4. Let g and h be positive continuous functions on the interval [a, b] such that h is increasing and g be decreasing functions on the interval
Proof. Under the conditions stated in Theorem 4, we can write Multiplying both sides of (2.13) where F(x, t) is defined by (2.5), we get (2.14) Integrating (2.14) with respect to t over (a, x), we have Again, multiplying (2.16) by and integrating the resultant identity with respect to ρ over (a, x), we get which completes the desired inequality (2.11) of Theorem 4.
Proof. Multiplying (2.16) by is defined by (2.5)) and integrating the resultant identity with respect to ρ over (a, x), we get

Dividing both sides by
which gives the desired inequality (2.32).
Proof. By multiplying both sides of (2.23) by is defined by (2.5)) and integrating the resultant identity with respect to ρ over (a, x), we have Hence, dividing (2.25) by which completes the desired proof.
−h ϑ (ρ)g (where F(x, ρ) is defined by (2.5)) and integrating the resultant identity with respect to ρ over (a, x), we get which completes the desired inequality (2.26) of Theorem 8.