Some new integral inequalities via general forms of proportional fractional integral operators

1COMSATS University Islamabad, Lahore Campus, Pakistan 2Department of Mathematics, Faculty of Arts and Sciences, Ağrı İbrahim Çeçen University, Ağrı, Turkey 3Institute of Graduate Studies, Ağrı İbrahim Çeçen University, Ağrı, Turkey 4Department of Mathematics, Faculty of Science and Arts, Giresun University, Giresun, Turkey 5International Center for Basic and Applied Sciences, Jaipur, India 6Department of Mathematics, Anand International College of Engineering, Jaipur, India


Introduction and preliminaries
Fractional analysis has brought a new dimension to many fields in mathematics and has become one of the most popular topics in recent years with its applications in several disciplines such as engineering, physics, modeling and control theory. Researchers have started to work intensively on fractional integral and derivative operators, and many new concepts and new applications have been included in the literature. The new features added by each new operator tries to prove their effectiveness in the real world problems solutions and the adventure continues in the search for the most effective operators. Many studies were conducted with the help of these operators to explain physical phenomena and demonstrate wide usage area in inequality theory (see the papers [3,7,12,19,20,24,27,30,[34][35][36]). Now, we take a look at fractional integrals from a historical perspective by recalling these operators.
respectively. Here Γ( ) is the Gamma function and its definition is It is to be noted that J 0 Riemann-Liouville integral operators are presented as a generalization of classical integrals. Then a more general version of this useful operator is given as follows.
Definition 1.2. [22] Let (a, b) with −∞ < a < b < ∞ be a finite or infinite interval of the real line R and α a complex number with Re(α) > 0. Also let h be a strictly increasing function on (a, b), having a continuous derivative h on (a, b). The generalized left and right sided Riemann-Liouville fractional integrals of a function f with respect to another function h on In [15], Jarad et al. investigated the generalized proportional fractional integrals with a different kernel structure that satify several important properties as follows:
We will continue with the Hadamard integral operators and the Katugampola integral operators, which are a general variant of the Riemann-Liouville integral operators as follows: The associated integral operator based on the Hadamard derivative operator is given as follows.
Within the scope of fractional analysis studies, the search for the operator with the most effective and general kernels led the researchers to define the operator named generalized proportional Hadamard fractional integrals. This operator, which has a different structure, is given as follows.
Definition 1.6. [26] The left and right generalized proportional Hadamard fractional integrals of order α > 0 and proportionality index ζ ∈ (0, 1] is defined by On all of these, the generalized proportional fractional (GPF) integral operator in the sense of another function h has been defined as follows.
If we set the parameters with different choices in Definition 1.7, one can obtain Riemann-Liouville integrals, generalized Riemann-Liouville fractional integrals, generalized proportional fractional integrals, Katugampola fractional integrals, Hadamard fractional integrals and generalized proportional Hadamard fractional integrals. In [25], Pečarić et al. mentioned about some different classes of convex functions as followings: A function f : I → [0, ∞) is said to be log −convex or multiplicatively convex (AG−convex) if log f is convex, or, equivalently, for all x, y ∈ I and t ∈ [0, 1] one has the inequality: A function f : I → [0, ∞) is said to be GA−convex if for all x, y ∈ I and t ∈ [0, 1], one has the inequality: The researchers have performed numerous research articles on various integral inequalities by using different kinds of fractional integral operators with applications, see [1, 2, 4-6, 8-11, 13, 14, 16-18, 23, 28, 31-33].
The main aim of this paper is to establish some new integral inequalities for product of two geometrically convex functions via the general forms of proportional fractional integral operators.
Proof. By using the definition of AG-convex functions, we can write By changing of the variable such that x = ta + (1 − t)b, we have By applying the General Cauchy inequality to above inequality, we get .
By multiplying both sides of the resulting inequality with By applying the well known Hölder inequality, we get By making use of some necessary operation and by taking into account ψ (x) ≤ M , we obtain The proof is completed.
Proof. By using the definition of AG−convexity, we can write By changing of the variable, we get By applying General Cauchy inequality and by making some operations, we have .
By multiplying both sides of the above inequality we obtain By applying Hölder inequality to the resulting inequality, we get By using the boundedness of ψ (x) ≤ M and some further calculation, we have This is the desired result.
Proof. By using the definition of GA−convexity, we can write .
By making use of some arrangements, we have . .