Lemniscate and Exponential Starlikeness of Regular Coulomb Wave Functions

Theorem 1. [24, Theorem 2.3b, p.28] Let ψ ∈Ψn[Ω, q] with q(0) = a. If p ∈H[a, n] satisfies ψ(p(z), zp′(z), z2p′′(z)) ∈Ω then p(z) ≺q(z). In [24], the authors discussed the class of admissible functions Ψ[Ω, q] when the function q maps D onto a disk or a half-plane. Very recently, taking the subordinate function q(z) = √1 + z Madaan et al. [21] discussed the admissible function class Ψ[Ω, √1 + z] and gave a particular case of the Theorem(1) as follow:

Lemma 1. [21] Let p ∈H[1, n] with p(z) ̸≡1 and n ≥1. Let Ω⊂C and Ψ : C3 ×D →C satisfies the admissibility condition Ψ(r, s, t; z) /∈Ωwhenever z ∈D, for r = √ 2 cos 2θeiθ, s = me3iθ 2 √ 2 cos 2θ and Re  1 + t s  ≥3m

Lemma 2. [26] Let Ωbe a subset of C and the function Ψ : C3 × D →C satisfies the admissibility condition Ψ(r, s, t; z) /∈Ωwhenever r = eeiθ, s = meiθr and Re  1 + t s  ≥m(1 + cos θ) where z ∈D, θ ∈[0, 2π) and m ≥1. If p is analytic function in D with p(0) = 1 and Ψ p(z), zp′(z), z2p′′(z); z
∈Ω for z ∈D, then p(z) ≺ez.

Theorem 2. Let η, L ∈C. If ( √ 2 −1) |2L −1| + 2 |η| < √ 2 4 , then the normalized regular Coulomb wave function z 7→gL,η(z) is lemniscate starlike in the unit disk D.

Theorem 3. Let η, L ∈C. If (e −1) |2L −1| + 2 |η| < e −1 e2 , then the normalized regular Coulomb wave function z 7→gL,η(z) is exponential starlike in the unit disk D.