On Geometric Properties of Bessel–Struve Kernel Functions in Unit Disc

: The Bessel–Struve kernel function deﬁned in the unit disc is used in this study. The Bessel–Struve kernel functions are generalized in this article, and differential equations are derived. We found conditions under which the generalized Bessel–Struve function is Lemniscate convex by using a subordination technique. The relation between the Janowski class and exponential class is also derived.


Introduction
This article focused on the Bessel-Struve Kernel function. This study explores a range of possible geometric features, including Lemniscate and exponential Carathéodory properties, and Lemniscate convexity. The details of these particular functions, as well as the geometric properties required, are explained further below.

Bessel-Struve Kernel Functions
Consider the Bessel-Struve kernel function B ν defined on the unit disk D = {z : |z| < 1} as where j ν (z) := 2 ν z −ν Γ(ν + 1)J ν (z) and h ν (z) := 2 ν z −ν Γ(ν + 1)H ν (z) are, respectively, known as the normalized Bessel functions and the normalized Struve functions of the first kind of index ν. More information about the Bessel and Struve functions can be found in [1,2]. The Bessel-Struve transformation and the Bessel-Struve kernel functions have appeared in many articles [3][4][5][6][7]. In [6], Hamem et al. studied an analog of the Cowling-Price theorem for the Bessel-Struve transform defined on a real domain and also provide Hardy's type theorem associated with this transform. The Bessel-Struve intertwining operator on C is considered in [4], and R is studied in [7]. The Fock space of the Bessel-Struve kernel functions is discussed in [5]. The monotonicity and log-convexity properties for the Bessel-Struve kernel and the ratio of the Bessel-Struve kernel and the Kummer confluent hypergeometric function are investigated in [3]. The kernel z → B ν (λz), ν ∈ C is the unique solution of the initial value problem L ν u(z) = λ 2 u(z), u(0) = 1, u (0) = νΓ(ν + 1) √ πΓ(ν + 3 2 ) . (2) L ν (u(z)) := d 2 u dz 2 (z) + Now, the Bessel functions and the Struve functions of order ν, respectively, have the power series This implies that B ν (taking λ = 1) possesses the power series The kernel B ν also have the integral representation It is evident from (2) and (3) that B ν satisfies the differential equation where . From (3) a computation yields that B ν satisfies the recurrence relation This article considers the function defined by Here, J ν,b,c is the Generalized Bessel function and S ν,b,c is the Generalized Struve function. A detailed study about the function J ν,b,c can be seen in the book [8], while the function S ν,b,c was first studied in [9]. There have been several articles where geometric properties such as close-to-convexity, starlikeness and convexity, radius of starlikeness and convexity of Bessel and Struve functions, along with their generalizations, were studied [9][10][11][12][13][14][15][16][17][18][19].
More development and properties about the Generalized Bessel-Struve kernel function B ν,b,c along with the differential equation is discussed in Section 2. More specifically, the power series of B ν,b,c is established, and it is shown that B ν,b,c is a solution of a secondorder differential equation.
Section 3 is devoted to the study of the geometric properties B ν,b,c . In particular, we derived the conditions on parameters ν, b, c for which B ν,b,c belongs to specific classes of geometric function theory, namely Lemniscate, Exponential and Janowski class. Detailed notes about geometric classes and terminologies are given below.

Basic Concept of Geometric Properties and Require Lemmas
Let A denote the class of normalized analytic functions f in the open unit disk D = {z : |z| < 1} satisfying f (0) = 0 = f (0) − 1. Denote by S * and C, respectively, the widely studied subclasses of A consisting of univalent (one-to-one) starlike and convex functions. Geometrically, f ∈ S * if the linear segment tw, 0 ≤ t ≤ 1, lies com- is a convex domain. Related to these subclasses is the Cárathèodory class P consisting of analytic functions p satisfying p(0) = 1 and Re p(z) > 0 in D.
For two analytic functions f and g in D, the function f is subordinate to g, written f ≺ g, or f (z) ≺ g(z), z ∈ D, if there is an analytic self-map ω of D satisfying ω(0) = 0 and f (z) = g(ω(z)), z ∈ D.
The class of Janowski starlike functions S * [A, B] consists of f ∈ A satisfying These are all classes that have been widely studied; see, for example, in the works of [20][21][22].
The next important class is related to the right half of the lemniscate of Bernoulli given by w : |w 2 − 1| = 1 . The functions p(z) = 1 + c 1 z + · · · in D satisfying p(z) ≺ √ 1 + z are known as lemniscate Cárathèodory function, and the corresponding class is denoted by P L . A lemniscate Cárathèodory function is also a Cárathèodory function and, hence, univalent. The class S L , known as lemniscate starlike, consists of functions f ∈ A such that The third important class that is considered in the sequel relates to the exponential functions e z . The functions p(z) = 1 + c 1 z + · · · in D satisfying p(z) ≺ e z are known as exponential Cárathèodory function, and the corresponding class is denoted by P E . The class S E , known as exponential starlike, consists of functions The principle of differential subordination [23,24] provides an important tool in the investigation of various classes of analytic functions. The following results are useful in a sequel.

Generalization of Bessel-Struve Kernel Function
To discuss the structure of Generalized Bessel-Struve kernel function along with various properties, lets recall about the Generalized Bessel function J ν,b,c from the article [8] and Generalized Struve function S ν,b,c from [9].
The functions J ν,b,c and S ν,b,c are, respectively, solutions of the differential equation and Both functions have the power series representation as follows where κ = ν + (b + 1)/2. The next result is about the power series of the Generalized Bessel-Struve kernel functions.

Proposition 1 (Power Series).
For ν > −1/2, the generalized Bessel-Struve functions have the power series of the form Proof. From the definition (8) of B ν,b,c , it follows that The Legendre duplication formula (see [1,2]) shows that Using these identities and the arrangement of odd and even terms, (16) can be rewritten as This complete the proof.
Proposition 2 (Differential Equations). The generalized Bessel-Struve function B ν,b,c is the solution of the differential eqaution Proof. In search of the series solution of (17), consider F(z) = ∑ ∞ n=0 A n z n the solution of (17). From the second differentiation and by arrangement of terms, it follows that Comparing the coefficients, we have This gives the odd coefficients as follows: and continuing this way, the odd coefficients have the general form Similarly, the odd coefficients can be determined as follows: and continuing like this, the general form of even terms are as follows: Finally, by considering A 0 = 1, the series solution is This completes the proof.

Relation with Lemniscate Class
This section finds the conditions on the parameters of the generalized Bessel-Struve kernel functions B ν,b,c (z) for which it is Lemniscate Carathéodory and convex in the unit disc. The first result finds the condition on ν, b, c for which B ν,b,c (z) ≺ √ 1 + z, while the second result discusses 1 + zB ν,b,c (z)/B ν,b,c (z) ≺ √ 1 + z.

Relation with Exponential Class
In this part, we derive sufficient conditions on L and η for which f ν (z) ≺ e z .

Relation with Janowski Class
In this section, we shall discuss the relation of generalized Bessel-Struve kernel functions with the Janowski class P [A, B]. Consider any one of the following and (iv) For B > 0, Proof. Define the analytic function p : D → C by where f ν (z) = B ν,b,c (z). Then, a computation yields and Using the Identities (32)-(34), the Bessel differential Equation (17) can be rewritten as Assume Ω = {0}, and define Ψ(r, s, t; z) by The Equation (35) yields that Ψ(p(z), zp (z), z 2 p (z); z) ∈ Ω. To ensure Re p(z) > 0 for z ∈ D, we will use the Lemma 1. Hence, it suffices to establish Re Ψ(iρ, σ, µ + iν; z) ≤ 0 in D for real ρ, σ such that σ ≤ −(1 + ρ 2 )/2, and σ + µ ≤ 0. Applying those inequalities we obtain The proof will be divided into four cases. Consider first B = −1, A > 3 − 2 √ 2. According to (37), we have We note that the function H is even with respect to ρ, and that satisfies H(0) ≤ 0, by virtue of an inequality in (28) along with the fact that with H (ρ) = 0 if and only if ρ = 0 or We observe that ρ 2 0 > 0 trivially for A = 1, and for A < 1, it holds by the inequality which is true due to the right side inequality given in (28). Further, in view of (39). Hence, H(ρ 0 ) = H max (ρ), and In the second case, we consider The inequality (37) reduces then to the following Re Ψ(iρ, σ, µ + iν; z) Clearly the quadratic function G is nonpositive for any ρ ∈ R, if  for any real ρ. Thus Taking A = 1 and B = −1 in Theorem 4 gives the following result Corollary 3. Re(B ν,b,c (z)) > 0 for √ π Re(k − 1)Γ k − 1 2 ≥ 2 √ c Γ(k). In particular, for ν ∈ R, Re(B ν,1,1 (z)) > 0 when ν > 1.5.

Concluding Remarks and Future Problems
By applying Lemma 2, we are able to drive the criteria for the convexity of generalized Bessel-Struve kernel functions B ν,b,c (z) in the lemniscate domain. The exponential convexity and Janowski convexity, however, cannot be produced in the same way. Using (26) and applying Lemma 3, we attempt to derive conditions on κ. However, there is no feasible κ for which the Lemma 3 assumptions are satisfied. Using Lemma 1, one can make a similar observation that the relationship with the Janowski Convex or convex with B ν,b,c (z) is not possible. Thus, further theoretical concepts or different approaches require studying the exponential or Janowski convexity or convexity of B ν,b,c (z).

Funding:
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
Institutional Review Board Statement: Not Applicable.