Connections between normalized Wright functions with families of analytic functions with negative coefficients

1 Introduction

Let 𝒜 denote the set all analytic normalized functions in the unit disc 𝒰 := { z ∈ ℂ : | z | < 1 } , i.e., such that f ⁢ ( 0 ) = 0 and f ′ ⁢ ( 0 ) = 1 . Therefore f ∈ 𝒜 has the form

Also, let 𝒮 denote the subset of 𝒜 consisting of functions which are univalent in 𝒰 .

We denote these classes by 𝒮 * ⁢ ( α ) and 𝒦 ⁢ ( α ) , respectively. Note that 𝒮 * := 𝒮 * ⁢ ( 0 ) and 𝒦 := 𝒦 ⁢ ( 0 ) , where 𝒮 * and 𝒦 are the classes of starlike and convex functions, respectively (for details, see [3], [20], [21] and [24]).

Kulkarni in [9] introduced a subset of 𝒮 , say 𝒟 ⁢ ( α , β , γ ) , defined as follows:

The following lemma, due to [9]), is essential for our results:

It is well known that special functions play an significant role in geometric function theory. Several special functions, called recently special functions of fractional calculus, play a very important and interesting role as solutions of fractional order differential equations, such as the Mittag-Leffler function, Wright function with its auxiliary functions, and Fox’s H-function (refer to [4] and [12] for a comprehensive treatment.)

Wright function plays an important role in the solution of linear partial fractional differential equations; in fact to solve the boundary-value problems for the fractional diffusion-wave equation, i.e., the linear partial integro-differential equation obtained from the classical diffusion or wave equation by replacing the first or second-order time derivative by a fractional derivative of order α with 0 < α ≤ 2 , it known that the corresponding Green functions can indeed be represented in terms of the Wright function (these aspects are presented in [5] and [7]).

We recall, for completeness, the definition of Wright function W λ , μ ⁢ ( z ) introduced by Wright in [25]:

Series (1.2) is absolutely convergent in ℂ when λ > - 1 and absolutely convergent in the open unit disc for λ = - 1 . Also, for λ > - 1 , the Wright function is an entire function.

If λ is a positive rational number, then W λ , μ ⁢ ( z ) can be represented in terms of the more familiar generalized hypergeometric functions (see [5, Section 2.1]). In particular, when λ = 1 and μ = p + 1 , the functions W 1 , p + 1 ⁢ ( - z 2 4 ) are expressed in terms of the Bessel functions J p ⁢ ( z ) as

We point out that W λ , p + 1 ⁢ ( - z ) ≡ J p λ ⁢ ( z ) ( λ > 0 , p > - 1 ) is known as the generalized Bessel function (sometimes it is also called the Bessel–Wright function). We refer to [5] for a comprehensive exposition of Wright’s functions.

Many researchers have investigated classes of analytic functions involving special function F ⊂ 𝒜 , to find different conditions such that the members of F have certain geometric properties such as starlikeness and convexity in 𝒰 . There is widespread literature dealing with various properties, generalizations and applications of different types of hypergeometric functions, especially for the generalized Gaussian, Kummer hypergeometric and Bessel functions (see [1], [10] and [15, 16, 17, 18]).

Note that W λ , μ ⁢ ( z ) as defined by (1.2) does not belong to the class T. Thus it is natural to consider the following normalizations of the Wright function:

and

From this, we can easily write

and

Furthermore, we observe that the normalized Wright functions W ( 1 ) ⁢ ( λ , μ ; z ) and W ( 2 ) ⁢ ( λ , μ ; z ) verify the following relations:

In [22] Silverman studied starlikeness and convexity properties for hypergeometric functions, giving necessary and sufficient conditions for hypergeometric functions F ⁢ ( a , b ; c ; z ) to be in the classes S * ⁢ ( α ) and C ⁢ ( α ) , examining also a linear operator acting on hypergeometric functions. Geometric properties of the normalized Wright functions were studied extensively by various authors like Baricz, Toklu and Kadioğlu [2], El-Shahed and Salem [4], Kilbas, Saigo and Trujillo [7], Mustafa [11], Mustafa and Altintas [12] and Prajapat [19]. In fact, the more generalized Fox–Wright hypergeometric functions were studied by Srivastava [23].

The contribution of the present article consists in establishing sufficient conditions so that the normalized Wright functions W ( 1 ) ⁢ ( λ , μ ; z ) and W ( 2 ) ⁢ ( λ , μ ; z ) belong to the class 𝒫 * ⁢ ( α , β , γ ) . Furthermore, we have obtained sufficient conditions for the integral transforms involving the normalized Wright functions W ( 1 ) ⁢ ( λ , μ ; z ) and W ( 2 ) ⁢ ( λ , μ ; z ) belong to 𝒫 * ⁢ ( α , β , γ ) .

2 Main results

Unless otherwise mentioned, we assume throughout this paper that μ > 0 , λ > - 1 , 0 < β ≤ 1 , 0 ≤ α < 1 2 ⁢ γ and 1 2 ≤ γ ≤ 1 . Moreover, we recall the Pochhammer symbol ( μ ) n = Γ ⁢ ( μ + n ) Γ ⁢ ( n ) = μ ⁢ ( μ + 1 ) ⁢ ( μ + 2 ) ⁢ ⋯ ⁢ ( μ + n - 1 ) , ( μ ) 0 = 1 defined in terms of the Euler gamma function.

3 Sufficient conditions for the integrals involving normalized Wright functions

In recent years several interesting families of integral operators have been investigated rather extensively studied by various authors (see for example [13, 14]) in analytic function theory, including each of the following integral operators:

and

where the function f ⁢ ( z ) belongs to the class 𝒜 and the parameters p , q are complex numbers such that the integrals in (3.1) and (3.2) exist.

In this section we establish sufficient conditions for the integral operators of type (3.1) and (3.2) when the function f ⁢ ( z ) is the normalized Wright functions to be univalent in the open unit disc U.

Let

and

where W ( 1 ) ⁢ ( λ , μ ; z ) and W ( 2 ) ⁢ ( λ , μ ; z ) are defined by (1.3) and (1.4), respectively. Note that G 1 ⁢ ( λ , μ ; z ) and G 2 ⁢ ( λ , μ ; z ) are in the class T.

By setting γ = 1 in Theorem 3 and Theorem 4, we obtain the following corollaries.