Radii of starlikeness and convexity of regular Coulomb wave functions

Lemma 1. If λ ∈[0, 1], a > b > 0 and z ∈C such that |z| < b, then (2.1) λ Re z2 a(a ± z) −Re z2 b(b ± z) ≥λ |z|2 a(a ± |z|) − |z|2 b(b ± |z|). Now, we are going to determine the radii of starlikeness of the normalized Coulomb wave functions.

Theorem 1. Let L > −1, η ≤0 and β ∈[0, 1). Then the radius of starlikeness of order β of the function fL,η is the smallest positive root of the equation rF ′ L,η(r) −β(L + 1)FL,η(r) = 0. Moreover, the radius of starlikeness of order β of the function gL,η is the smallest positive root of the equation (L + β)FL,η(r) −rF ′ L,η(r) = 0. It is important to mention that the regular Coulomb wave function is actually a generalization of a transformation of the Bessel function of the first kind. Namely, w

Corollary 1. If L > −1 2 and β ∈[0, 1), then the radius of starlikeness of order β of the function z 7→fL−1 2 ,0(z) =  2LΓ(L + 1)√zJL(z)  1 L+ 1 2 is the smallest positive root of the equation rJ′ L(r) −  β

Theorem 2. If L > −1 and η ≤0, then the radii of univalence of the functions fL,η and gL,η correspond to the radii of starlikeness. Moreover, if L > −1 and η < 0, then we have that (2.2) r∗(fL,η) > (L + 1)2√2L + 3 p L4 + 6L3 + (η2 + 12)L2 + 2(3η2 + 5)L + 3(2η2 + 1)

Theorem 3. If L > −1 2 and η ≤0, then the radius of convexity of order β ∈[0, 1) of fL,η is the smallest positive root of the equation (2.6) 1 + rF ′′ L,η(r) F ′ L,η(r) − L L + 1 rF ′ L,η(r) FL,η(r) = β. Moreover, if L > −1 and η ≤0, then the radius of convexity of order β of gL,η is the smallest positive root of the equation

Corollary 2. If L > 0 and β ∈[0, 1), then the radius of convexity of order β of the function fL−1 2 ,0 is the smallest positive root of the transcendental equation 1 + r2J′′ L(r) + rJ′ L(r) −1 4JL(r) rJ′ L(r) + 1 2JL(r) − L L + 1 rJ′ L(r) + 1

Lemma 2. Let Ω⊆C be a set in the complex plane C and Ψ : C3 × D →C a function, that satisfies the admissibility condition Ψ (ρi, σ, µ + iν; z) /∈Ω, where z ∈D, ρ, σ, µ, ν ∈R with µ + σ ≤0 and σ ≤−(1+ρ2)/2. If h is analytic in the unit disk D, with h(0) = 1 and Ψ h(z), zh′(z), z2h′′(z); z
∈Ωfor all z ∈D, then Re h(z) > 0 for all z ∈D. In particular, if we only have Ψ : C2 × D →C, the admissibility condition reduces to Ψ (ρi, σ; z) /∈Ωfor all z ∈D and ρ, σ ∈R with σ ≤−(1 + ρ2)/2. Now, we are rea

Theorem 4. If η, L ∈C are such that Re L ≥1 2, Im L ≥1 and (1 + Im L + |η|)2 ≤ Re L −1 2
2, then Re gL,η(z) > 0 for all z ∈D. Moreover, if |η| ≤Re L −1 3 (Im L)2 −1 4, then gL,η map the open unit disk D into a starlike domain with respect to origin.

Theorem 3 we used the fact that for L > −1 and η ∈R the function r 7→rF ′ L,η(r) −LFL,η(r) has only real zeros. Now we are asking what happens in general with the zeros of the linear combinations of the normalized Coulomb wave function and its derivative. Open Problem 3. Under which conditions on L, η and α the function r 7→rF ′ L,η(r) + αFL,η(r) has only real zeros? In the proof of the main results concerning the radii of univalence, starlikeness and convexity it was essential the fact that η ≤

Lemma 2 we conclude that Re qL,η(z) > 0 for all z ∈D, which show that gL,η is starlike in D. □ References [1] I. Aktas¸, ´A. Baricz, N. Ya˘gmur, Bounds for the radii of univalence of some special functions, (submitted). [2] ´A. Baricz, Tur´an type inequalities for regular Coulomb wave functions, J. Math. Anal. Appl. 430 (2015) 166–180. [3] ´A. Baricz, M. C¸a˘glar, E. Deniz, Starlikeness of Bessel functions and their derivatives, Math. Inequal. Appl. 19(2) (2016) 439–449. [4] ´A. Baricz, E. Deniz