Abstract
In this paper our aim is to determine the radii of univalence, starlikeness and convexity of the normalized regular Coulomb wave functions for two different kinds of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for regular Coulomb wave functions, and properties of zeros of the regular Coulomb wave functions and their derivatives. Moreover, by using the technique of differential subordinations we present some conditions on the parameters of the re
Results & Lemmas (10)
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Lemma 1.
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 ±…
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. · radius
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…
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. · radius
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) …
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.
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.…
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. · radius
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…
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. · radius
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…
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.
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;…
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.
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,…
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
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…
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 · radius
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.…
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
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