On the Janowski Starlikeness of the Coulomb Wave Functions

Let A be the class of functions f which are analytic in the open unit discU = fz : jzj < 1g and normalized by the conditions f ð0Þ = f ′ð0Þ − 1 = 0: An analytic function f is subordinate to an analytic function g (written as f ≺ gÞ if there exists an analytic function w with wð0Þ = 0 and jwðzÞj < 1 for z ∈U such that f ðzÞ = gðwðzÞÞ: In particular, if g is univalent in U, then f ð0Þ = gð0Þ and f ðUÞ ⊂ gðUÞ: Let P 1⁄2A, B denote the class of analytic functions p such that pð0Þ = 1 and


Introduction
Let A be the class of functions f which are analytic in the open unit disc U = fz : jzj < 1g and normalized by the conditions f ð0Þ = f ′ ð0Þ − 1 = 0: An analytic function f is subordinate to an analytic function g (written as f ≺ gÞ if there exists an analytic function w with wð0Þ = 0 and jwðzÞj < 1 for z ∈ U such that f ðzÞ = gðwðzÞÞ: In particular, if g is univalent in U, then f ð0Þ = gð0Þ and f ðUÞ ⊂ gðUÞ: Let P ½A, B denote the class of analytic functions p such that pð0Þ = 1 and Note that for 0 ≤ β < 1,P ½1 − 2β,−1 is the class of analytic functions p with pð0Þ = 1 satisfying Re pðzÞ > β in U: For −1 ≤ B < A ≤ 1, the class S * ½A, B defined by is the class of Janowski starlike functions [13]. For 0 ≤ β < 1, S * ½1 − 2β,−1 ≔ S * ðβÞ is the usual class of starlike functions of order β: These classes have been studied in [6,8]. A function f ∈ A is said to be close-to-convex of order β with respect to a function g ∈ S * if Re ðzf ′ ðzÞ/gðzÞÞ > β: In particular case, if f ∈ A and satisfies the condition Re f ′ðzÞ > β for all z in U, then f ðzÞ is a close-to-convex of order β.
Let 1 F 1 denote the Kummer confluent hypergeometric function. The regular Coulomb wave function is defined as where , a L,n = 2ηa L,n−1 − a L,n−2 n n + 2L which is the solution of following differential equation: In this paper, we focus on the following normalized form: The function g L,η ðzÞ satisfies the following homogenous second-order differential equation: Baricz [9,10] studied the Turan-type inequalites of regular Coulomb wave functions and zeros of a cross-product of the Coulomb wave and Tricomi hypergeometric functions, respectively. Baricz et al. [11] also investigated the radii of starlikeness and convexity of regular Coulomb wave functions. Recently, Aktas [1] has studied lemniscate and exponential starlikeness of Coulomb wave functions. In some recent papers [2][3][4][5]12], the authors have discussed certain geometric properties of some special functions. The relationships of generalized Bessel function, Bessel-Struve kernal function, and Struve function with the Janowski class have also been studied by various researchers, see [7,14,17,18]. Motivated by the above papers in this subject, in this paper, our aim is to present some geometric results for the normalized regular Coulomb wave function.
The following lemmas are needed in the paper.

Inclusion of Generalized Coulomb Wave Function in the Janowski Class
Our first result is related with Janowski starlikeness of normalized Coulomb wave function.
Proof. Define an analytic function p : U ⟶ ℂ by Then, Journal of Function Spaces A rearrangement of (18) gives.
Now, define a function q L,η : U ⟶ ℂ by This function q L,η is analytic in U and q L,η ð0Þ = 1: Suppose that z ≠ 0: We know that g L,η ðzÞ ≠ 0: This function satisfies the following equation: which yields Substituting (17) and (19) in (22), we get or equivalently Now setting Then, for ρ ∈ ℝand σ = −ð1 + ρ 2 Þ/2, we get To get the contradiction, we have to show QðρÞ ≤ 0 for 3 Journal of Function Spaces ρ ∈ ℝ: We split the proof into two cases. First, consider the case B = −1 < A ≤ 1: Then, the function Q becomes that achieve its maximum at ρ 0 = −2 Im ðLÞ/ð2 + AÞ, and which is nonpositive if and only if Now, consider the case −1 < B < A ≤ 1: Rewriting Q in the form where The inequality QðρÞ ≤ 0 holds for any real ρ, if P > 0, S > 0 and R 2 ≤ 4PS or that holds by hypothesis (14) and (15). Thus, in both cases, the function Ψ satisfies the hypothesis of lemma (8) and hence Re pðzÞ > 0, or By definition of subordination, there exist a map ω in Uwith ωð0Þ = 0, and which yields Hence, If take A = 1 − 2β and B = −1 for 0 ≤ β < 1 in Theorem 3, we obtain following result.