Certain Geometric Properties of the Fox–Wright Functions

: The primary objective of this study is to establish necessary conditions so that the normalized Fox–Wright functions possess certain geometric properties, such as convexity and pre-starlikeness. In addition, we present a linear operator associated with the Fox–Wright functions and discuss its k -uniform convexity and k -uniform starlikeness. Furthermore, some sufﬁcient conditions were obtained so that this function belongs to the Hardy spaces. The results of this work are presumably new and illustrated by several consequences, remarks, and examples


Introduction and Motivation
In this article, we consider the Fox-Wright function; we studied several geometric properties in a unit disc.The Fox-Wright function is an important special function that plays a vital role in different branches of science and engineering.Geometric properties, such as starlikeness, pre-starlikeness, convexity, k-uniformly convexity, and k-uniformly starlikeness, are associated with special functions, and have always been focused on by researchers [1][2][3][4].For more descriptions of the geometric properties of special functions, please refer to [1][2][3][4][5][6][7][8][9][10] and their references.
The results mentioned above motivated us to discuss the geometric properties related to Fox-Wright functions, such as pre-starlikeness, convex of order δ, k-uniform starlikeness, k-Uniform convexity, and Hardy spaces.
In addition to explaining the definitions of the geometric terms mentioned above, we will now introduce the basic definitions and properties of the Fox-Wright function, which are important in the sequel.

Geometric Functions Theory
Suppose A is the class of analytic functions of the form is univalent in D and the starlike domain with respect to the origin in C. The class of starlike functions in D is denoted by S * .The analytical descriptions of starlike functions (see [15]) can be described as: Let S * (α) denote the starlike function of order α.Then analytical characterization of S * (α) is given by: zg (z) is a convex domain in C. Suppose that K denotes the class of convex functions.The analytical description of K is given by Moreover, assume that K(α) denotes the class of convex functions of order α.Then the analytical characterization of K(α) is given as: It is well known that every close-to-convex in D is univalent in D.
The convolution (Hadamard product) of two power series is an important tool in the geometric functions theory.To define convolution (Hadamard product), let us consider the following two Taylor series: where ρ 1 and ρ 2 are their radii of convergence, respectively.Then their convolutions (Hadamard product), see, for example, [16], are represented as: We also require the convolution of two classes of analytic functions.If M 1 and M 2 consist of analytic functions defined on the unit disc, then Pre-starlike functions are introduced by Ruscheweyh in [17].For µ ∈ (0, 1], consider the function Then the class of pre-starlike functions of order µ, denoted by L µ , is given as L µ := g : g ∈ A and g µ (z) * g ∈ S * (µ) .
In particular, L 1 2 = S * 1 2 and L 0 = C. Sheil-Small et al. [18] generalized the class L µ as L[ρ, µ] for 0 ≤ ρ, µ < 1, which is defined as If a real function g ∈ A on (−1, 1) satisfies the following relation: then it is called a typical real function.In [19], Robertson introduced the class of typically real functions.A function g ∈ A is said to be convex in the imaginary axis direction, if the region g(D) is a convex region in the imaginary axis direction, i.e., It is well known from [19] that a function g ∈ A is convex in the imaginary axis direction containing real coefficients if zg (z) is a typically real function, which is equivalent to Two other important classes required in our study are k-ST and k-UCV, introduced by Kanas et al. (see [20] and [21]; see also [22]); they are defined as follows: k-ST = g : g ∈ S and zg (z) The following two results are significant for k-ST and k-UCV, respectively.
We also consider the following class: If we take the function g ∈ A in the class R η (ζ), then we have

Hardy Space
Let us consider the space of all bounded functions H ∞ in open unit disk D. We suppose that h is in the class of analytic functions in domain D and set It can be observed from [23] that h ∈ H q , if M q ( , h) is bounded for all ∈ [0, 1) and Let us consider the following results [24] related to the Hardy space H q : This paper is organized as follows.In Section 2, we present some lemmas that will help derive the main results.Section 3 presents sufficient conditions so that the Fox-Wright function satisfies certain geometric properties, such as pre-starlikeness and convexity of order δ.Furthermore, consequences, important remarks, and examples are shown in this section.In Section 4, we consider the linear operator associated with the Fox-Wright function for which k-uniformly starlikeness and k-uniformly convexity are discussed.Furthermore, sufficient conditions are established in Section 5, so that this function belongs to the Hardy space.Consequences and remarks are also presented in this section.

Useful Lemmas
Some Lemmas were recalled in this section, which will be helpful to prove the main theorems in this paper.
where z ∈ C .

Lemma 4 ([28]
). Suppose that the function h(z) ∈ A and Then h is starlike in D.
Let M be the class of all analytic functions in D and µ < 1.In [30], the following classes are introduced: For η = 0, we obtain the classes of the analytic function M o (µ) and L o (µ), respectively.
The following lemmas are required to prove the main results in Section 5.
The value of µ cannot be improved.
Lemma 8 ([33]).If the function g, convex of order λ (0 ≤ λ < 1), is not of the following type: for n, r ∈ C and for ζ ∈ R, then each of the following statements holds true: (i) There exists ρ = ρ(g), such that h ∈ H Now we are ready to state and prove the main results in the subsequent sections.

Pre-Starlikeness and Convexity
We should note that We consider the following normalized form of the Fox-Wright function: Proof.Let us consider the function p(z) in form of the Hadamard product, defined as To show the results stated by Theorem 3, it suffices to prove that From (6), we have where Let us define the function l 1 (s) as: Using the logarithmic derivative on (9), we have where Using Lemma 1, we have This shows that l 3 (s) is decreasing on [1, ∞) under the given hypothesis.It can also be verified that l 3 (1) < 0 and l 2 (s) < 0, for s ≥ 1.Consequently, l 1 (s) < 0 on [1, ∞).Clearly, (q k ) k≥1 is decreasing.With the help of (7), we have A simple computation leads to where the sequence (r k ) k≥1 is given by Similarly, it can be proven that (r k ) is decreasing.Now, using (11), we obtain Combining ( 7) and (11), we have which is equivalent to the given condition, i.e., Hence, the theorem is proved. .
Then the Fox-Wright function m ψm [.] is convex of order δ in D. Proof.Let To show the desired results, we have to prove that h(z) = zg (z) is a starlike function of order δ in D. For this, it is enough to prove that We have where Let us consider the function h 1 (s) defined as: Applying the logarithmic derivative on both sides, we have where Using Lemma 1, we have Therefore, This implies that h 3 is the decreasing function on [1, ∞) under the given hypothesis.It can also be observed that h 3 (1) < 0 and h 2 (s) < 0, for s ≥ 1.Consequently, h 1 (s) < 0 on [1, ∞).Thus, (c k ) k≥1 is a decreasing sequence.With the help of ( 13), we have By a simple computation, we have where (d k ) k≥1 is given by In view of sequence (c k ), it can be proven that (d k ) is decreasing.Now, using ( 16), we obtain Combining ( 15) and ( 17), we have which is equivalent to the given condition, i.e., Thus, proof of the theorem is completed.

k-Uniformly Starlike and k-Uniformly Convex Functions
In this section, we consider a linear operator associated with the Fox-Wright function for which k-ST and k-UCV are discussed.
Let us define a linear operator using the Hadamard product involving the normalized Fox-Wright function as: It can be noted that I m,n (g) (z) is a natural extension of the Alexander and Libra operators, denoted by A and L, respectively, which was first introduced in [35].It can be defined in terms of the Fox-Wright function as follows: . * g (z) and . * g (z).
These operators are very useful in fractional calculus.Applications of fractional derivatives involving the Alexander integral operator were discussed in [36].In [22], relations between k-UCV, k-ST, and R η (ζ) were studied for the above-mentioned similar type of linear operator associated with the hypergeometric function.Now, we establish some important theorems associated with the classes k-ST and k-UCV for the linear operator I m,n (g) (z).
Proof.To prove this theorem, it is enough to prove that where Let us consider the function h 1 (s) defined as: where Using Lemma 1, we have Which leads to It can be easily shown that h 3 (s) is a decreasing function on [1, ∞) under the given hypothesis.It can also noted that h 3 (1) < 0 and further h 2 (s) < 0, for s ≥ 1.Consequently, h 1 (s) < 0 on [1, ∞).Thus, (b n ) n≥1 is a decreasing sequence.Therefore, by (19), we obtain .
In view of condition (18), proof of this theorem is completed.Theorem 6.Let the given supposition of Theorem 4 hold with l ∈ [0, ∞) and f (z) ∈ R η (ζ).Moreover, assume that the following inequality holds: Proof.To find the required result, we have to show that A simple computation leads to where Using a similar technique proof of Theorem 4 and applying the assumption of this theorem, it can be observed that c n is a decreasing sequence.Given (22), we obtain Finally, using the given hypothesis, the desired result can be established.

Hardy Space of the Fox-Wright Function
In this section, we will study the inclusion properties of the Fox-Wright function in the Hardy space.Theorem 7.Under the same supposition and statement of Theorem 4, the following relation holds: Proof.Using the definition of the hypergeometric function 2 φ 1 (p, q; r; z) [13], we have It can be easily seen that the normalized Fox-Wright function m ψm (a 1 , b 1 ), . . ., (a m , b m ) (α 1 , β 1 ), . . ., (α n , β n ) z is not of the given types: Hence, by applying Theorem 4, we observe that m ψm (a α,β (z) is in the Hardy space for α ≥ 1.Therefore, our Theorem 3 improves the existing result in [3] and in the generalized form.

Theorem 8. Let the assertion of Theorem 4 hold along with the condition
Proof.To prove this theorem, it is enough to show that |q(z) − 1| < 1, where Now by using Lemma 1, we have By applying the similar way proof of Theorem 4, with the assumption of this Theorem, we have (e − 1).
Hence, followed by the given condition, the proof of this theorem is complete.
Theorem 9. Suppose that the same assumptions of Theorem 4 hold and the following inequality is satisfied: using the given conditions.Hence, by using Lemma 7, it can be noted that u ∈ M(ρ), and u ∈ L(µ).Thus, the proof of Theorem 9 would follow readily.

Conclusions
In our present article, we investigated geometric properties, such as convexity of order δ and pre-starlikeness for the Fox-Wright function m ψm [.].It can be observed from Remark 3 that some of the results obtained in this manuscript improved and generalized certain results established in [34].Moreover, sufficient conditions were derived, such as k-uniformly starlike and k-uniformly convex associated with this function.Many other conditions were also provided for the Fox-Wright function belonging to the Hardy space.From Remark 4 and Remark 5, we can observe that the results derived in this paper improved and generalized several results available in the literature [3,34].Interesting examples and consequences were provided to support the desired results obtained in this investigation.Further research directions on the subjects of the present considerations were discussed analogously, i.e., for the Fox-Wright type functions [37], hypergeometric function [13], and Srivastava's unification [38] E α,β (ϕ; z, s, κ) of the Mittag-Leffler type functions.

Theorem 5 .
Let the assertion of Theorem 4 hold and l
Finally, with the help of Lemma 8, the required result would readily follow.