On a new linear operator formulated by Airy functions in the open unit disk

In this note, we formulate a new linear operator given by Airy functions of the first type in a complex domain. We aim to study the operator in view of geometric function theory based on the subordination and superordination concepts. The new operator is suggested to define a class of normalized functions (the class of univalent functions) calling the Airy difference formula. As a result, the suggested difference formula joining the linear operator is modified to different classes of analytic functions in the open unit disk.


Introduction
The field of geometric function theory is rich with different types of linear, differential, integral, and mixed operators. A few linear operators have been formulated in this field, such as the Carlson-Shaffer operator [1], hypergeometric linear operator [2,3], and Fox-Write linear operator [4]. In this note, we present a linear operator formulated by the Airy functions [5], which are special functions determined by the hypergeometric function of a complex variable. These functions are solutions for the Airy equation f (z)zf (z) = 0.
The class of these differential equations plays an important role in applied sciences such as optics, economy, and astronomy. The greatest benefit of Airy functions in mathematical studies is development in the fields of special functions and statistical studies [6]. The formula of the Airy function of a complex variable is given by   (1/2) , and p q is the Fox-Wright function having the series Moreover, the Airy distribution function of the random variable χ is given by the formula (see Fig. 1) By using the complex probability [7,8], Eq. (2) can be extended to the complex domain as follows:

Methods
Let be the class of normalized functions in U having the series And let S * , C be the classes of starlike and convex functions respectively. The Hadamard product (convolution product) is defined by the series where g(z) = z + ∞ n=2 ψ n z n . An analytic function f ∈ U is on subordination with the analytic function g ∈ U represented by f ≺ g if there occurs an analytic function w with |w(z)| ≤ |z| such that f = (g(w)). In the sequel, we shall use the class of normalized functions satisfying f (0) = 0 and f (0) = 1 having the series (see [9]) Moreover, two analytic functions f and g in U, the function f is majored by g (f g) if there is an analytic function , | | < 1 such that f (z) = (z)g(z). Note that there is a connection between majorization and subordination concepts (see [10,11]). Under some conditions, we have f g ⇔ f ≺ g.

Linear operator
We shall use the Hadamard product to define the new linear operator using the Airy function of a complex variable z ∈ U. Construct the modified Airy function as follows: Define a linear operator : → as follows: where δ n indicates the coefficient of Aı(z). The linear operator (6) is called the Airy linear operator of normalized analytic functions. It is well known that for (z) > 0 the Airy function is convex with (Aı(z)) > 0. We have the following proposition, which indicates that the linear operator can be formulated by a set of special functions and other properties, which are easily proved. Therefore, we omit the proof.

The difference formula
We proceed to defining our class of normalized analytic functions based on the Airy equation. The Airy equation can be reformulated by the structure Our structure of the class of analytic functions is given by the Airy difference formula .
By utilizing the linear operator f , we have the following class.
Definition 1 Let f ∈ . Define the class of analytic functions ı s satisfying the following subordination: .
Moreover, we have It is clear that the formula And by comparing the coefficients of f (z) and s (z), we have that the unique real root of s 3 -3s 2 + 4s -1 = 0 is As a conclusion, we have Note that the function z (1-z) s is called the generalized Koebe function, which is an extreme function in U for some values of s.
Our investigation is based on the following result which can be located in [9].

Results
In the result section, we present the sufficient condition for functions to be in the class ı s .
Theorem 3 Let f ∈ , and for some constants s ∈ R \ {0} define the functional .
The function s (z) is analytic in U having s (0) = 1, and it is a solution of the differential equation 1 + s z s (z) s (z) = (1 + z) 1/2 , z ∈ U.