New Applications of the Fractional Integral on Analytic Functions

: The fractional integral is a function known for the elegant results obtained when introducing new operators; it has proved to have interesting applications. In the present paper, differential subordinations and superodinations for the fractional integral of the conﬂuent hypergeometric function introduced in a previously published paper are presented. A sandwich-type theorem at the end of the original part of the paper connects the outcomes of the studies done using the dual theories.


Introduction
It is a known fact that the notion of an operator was used from the early stage of the study of complex-valued functions; many already known results can be proven easier with them, and new results are being obtained with them.
Many papers, such as [1,2], studied different operators defined by using the fractional integral of order λ also used earlier by S. Owa [3]. Despite that, we also refer to [4][5][6] for theoretical and numerical analyses from real models described by classical PDEs and related operators. The results contained in the present paper were inspired by the outstanding results previously obtained using fractional integrals, and the study was done by applying them to a confluent hypergeometric function. The definition of a fractional integral can be seen in [3] as follows: Definition 1 ([3]). The fractional integral of order α is defined by where α is a positive real number, f (z) is an analytic function in a simply connected region of the z-plane containing the origin and the multiplicity of (ζ − z) α−1 is removed by requiring ln(ζ − z) to be real when (ζ − z) > 0.
In paper [7] a new operator was introduced by using a fractional integral on the confluent (Kummer) hypergeometric function. The introduction of this operator was inspired by the studies done on this function having in view many aspects, from its combination with other functions, as can be seen in papers [8,9], to its univalence in paper [10].
The confluent (Kummer) hypergeometric function of the first kind is defined in [11] as follows: . Let a, c ∈ C, c = 0, −1, −2, . . . and consider φ(a, c; z) = 1 F 1 (a, c; z) = 1 + a c z 1! + a(a + 1) c(c + 1) This function is called a confluent (Kummer) hypergeometric function, is analytic in C and satisfies Kummer's differential equation: the confluent (Kummer) hypergeometric function can be written as The definition of the operator introduced in [7] is the following: The fractional integral of a confluent hypergeometric function can be written as after a simple calculation. Evidently, D −λ z φ(a, c; z) ∈ H[0, λ]. The original results which are shown in the next part of this paper were obtained by using this operator and differential subordination and superodination theories, synthesized in the monography [12] published by Miller and Mocanu in 2000 and in paper [13], respectively. The usual notion and definitions are considered. U = {z ∈ C : |z| < 1} is the unit disc of the complex plane, H(U) the class of analytic functions in U and H[a, n] = { f ∈ H(U) : f (z) = a + a n z n + a n+1 z n+1 + . . . , z ∈ U}, with n a positive integer and a ∈ C. Definition 3 ([12]). Let f , F ∈ H(U). The function f is said to be subordinate to F if there exists a Schwarz function w, analytic in U, with w(0) = 0 and |w(z)| < 1, z ∈ U, such that f (z) = F(w(z)), z ∈ U. In such a case, we write f ≺ F. If F is univalent, then f ≺ F if and only if f (0) = g(0) and f (U) ⊂ g(U).

Definition 4 ([12]
). Let ψ : C 3 × U → C and let h be univalent in U. If p is analytic and satisfies the differential subordination then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant, if p ≺ q for all p satisfying (5). A dominant q that satisfies q ≺ q for all dominants q of (5) is said to be the best dominant of (5).
The notion related to differential superordinations was introduced in [13].

Definition 5 ([13]).
Let ϕ : C 3 × U → C and let h be analytic in U. If p and ϕ(p(z), zp (z), z 2 p (z); z) are univalent in U and satisfy the differential superordination then p is called a solution of the differential superordination (6). An analytic function q is called a subordinant of the solutions of the differential superordination or more simply a subordinant, if q ≺ p for all p satisfying (6). A subordinant q that satisfies q ≺ q for all subordinants q of (6) is said to be the best subordinant of (6).
In the process of obtaining the original results from this paper, the following lemmas are needed: 12]). Let the function q be univalent in the unit disc U and θ and φ be analytic in a domain D containing q(U) with φ(w) = 0 when w ∈ q(U). Set Q(z) = zq (z)φ(q(z)) and h(z) = θ(q(z)) + Q(z). Suppose that Q is star-like univalent in U and Re zh (z) then p(z) ≺ q(z) and q is the best dominant.

Lemma 2 ([14]
). Let the function q be convex univalent in the open unit disc U and ν and φ be analytic in a domain D containing q(U). Suppose that Re then q(z) ≺ p(z) and q is the best subordinant.
If the following subordination is satisfied by q, and q is the best dominant.

Theorem 2.
Let q be an analytic and univalent function in U with q(z) = 0, ∀ z ∈ U, such that zq (z)
Proof. Define the function p by , z ∈ U, z = 0. it is easy to show that φ is analytic in C\{0}, φ(w) = 0, w ∈ C\{0} and ν is also analytic in C. Since