New Applications of Fractional Integral for Introducing Subclasses of Analytic Functions

The fractional integral is prolific in giving rise to interesting outcomes when associated with different operators. For the study presented in this paper, the fractional integral is associated with the convolution product of multiplier transformation and the Ruscheweyh derivative. Using the operator obtained as a result of this association and inspired by previously published results obtained with similarly introduced operators, the class of analytic functions IR(μ,λ,β,γ,α,l,m,n) is defined and investigated concerning various characteristics such as distortion bounds, extreme points and radii of close-to-convexity, starlikeness and convexity for functions belonging to this class.


Introduction
The fractional integral was recently investigated in relation with many different functions. Interesting results were obtained when applying the fractional integral to a confluent hypergeometric function [1], in connection with Sȃlȃgean and Ruscheweyh operators [2], related to Bessel functions [3] or for the Mittag-Leffler Confluent Hypergeometric Function [4]. Applications of fractional calculus emerged in many studies related to convexity [5,6] and involving generalized fractional integral operators [7][8][9].
The applications of the fractional integral involved in the present study are related to a previously introduced operator obtained as a convolution product of a multiplier transformation and a Ruscheweyh derivative. In order to present the original results of the study, known notations and known definitions are used.
Consider H(U), the class of analytic function in U = {z ∈ C : |z| < 1}, the open unit disc of the complex plane, H(a, n), the subclass of H(U) of functions with the form f (z) = a + a n z n + a n+1 z n+1 + . . . and A n = { f ∈ H(U) : f (z) = z + a n+1 z n+1 + . . . , z ∈ U} with A = A 1 .
The Hadamard product (or convolution) of analytic functions in the open unit disc U, f (z) = z + ∑ ∞ k=2 a k z k and g(z) = z + ∑ ∞ k=2 b k z k , denoted by f * g, is defined as: The operators used for the present study are the following.
Inspired by the results seen in [17], a new subclass of analytic functions is defined using the operator given in Definition 5.

Coefficient Bounds
In this section, coefficient bounds and extreme points for functions in IR(µ, λ, β, γ, α, l, m, n) are obtained. Theorem 1. The function f ∈ A belongs to the class IR(µ, λ, β, γ, α, l, m, n) if and only if The result is sharp for the function Proof.
Assume that function f ∈ A and that Inequality (3) holds. Then we obtain: Choosing values of z on the real axis and considering z → 1 − , we have: Conversely, suppose that f ∈ IR(µ, λ, β, γ, α, l, m, n), then we obtain the following inequality: Taking account that Re(−e iθ ) ≥ −|e iθ | = −1, the above inequality reduces to: Letting r → 1 − and applying the mean value theorem, we have the desired inequality (3). This completes the proof of Theorem 1.

Theorem 4. The function f
The result is sharp, with the extremal function f given by (4).

Proof.
For the function f ∈ A, we have to show that: By a simple calculation we obtain Function f ∈ IR(µ, λ, β, γ, α, l, m, n) if and only if 1 Γ(n + 1) , which completes the proof.
Each of these results is sharp for the extremal function f given by (4).
2. The function f is convex if and only the function z f is starlike; therefore it is enough to prove (2) with a similar method as that of the proof of (1). Thus, the function f is convex if and only if: We obtain Function f ∈ IR(µ, λ, β, γ, α, l, m, n) if and only if 1 Γ(n + 1) and relation (16) is true if , which yields the convexity of the family.

Conclusions
The study presented in this paper followed the line of research regarding introducing new classes of univalent functions using different operators. The operator used for obtaining the original results of this paper is part of the celebrated family of fractional integral operators, much investigated in recent years. Using the operator presented in Definition 5, the new subclass of analytic functions under investigation in this paper, IR(µ, λ, β, γ, α, l, m, n), was introduced in Definition 6. The paper presented the results of the studies carried out on coefficients, for finding the distortion bound of the functions in the new class and for establishing domains of starlikeness, convexity and close-to-convexity for the functions in the class IR(µ, λ, β, γ, α, l, m, n) and finding radii associated with those domains.
As future lines of study involving the class IR(µ, λ, β, γ, α, l, m, n), aspects related to subordination and superordination properties could be investigated. Interesting results related to the relatively new concepts of fuzzy differential subordinations and superordinations might be also obtained.
The symmetry properties of the functions defined by an equation or inequality to obtain solutions with particular properties could be studied. The study of special functions by the differential subordinations method could give interesting results about their symmetry properties. In a future paper, the symmetry properties for different functions using the concept of quantum computing could be studied.