# On the Janowski Starlikeness of the Coulomb Wave Functions

Authors: Saddaf Noreen; Mohsan Raza; Erhan Deniz; Sarfraz Nawaz Malik
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we are interested in finding sufficient conditions onA,B,L, and η which ensure the normalized Coulomb wave function to be Janowski starlike. Sufficient conditions are also obtained for gL,η/z∈PA,B, which readily yield conditions for gL,η to be close-to-convex.


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## Body

## 1. Introduction

LetA be the class of functions f which are analytic in the open unit disc U=z:z<1 and normalized by the conditions f0=f′0−1=0. An analytic function f is subordinate to an analytic function g (written as f≺g if there exists an analytic function w with w0=0 and wz<1 for z∈U such that fz=gwz. In particular, if g is univalent in U,then f0=g0 and fU⊂gU. Let PA,B denote the class of analytic functions p such that p0=1 and
(1)pz≺1+Az1+Bz,−1≤B<A≤1,z∈U.Note that for0≤β<1,P1−2β,−1 is the class of analytic functions p with p0=1 satisfying Repz>β in U. For −1≤B<A≤1, the class S∗A,B defined by
(2)S∗A,B≔f∈A:zf′zfz≺1+Az1+Bz,z∈U,is the class of Janowski starlike functions [13]. For 0≤β<1,S∗1−2β,−1≔S∗β is the usual class of starlike functions of order β:
(3)S∗1−β,0≔Sβ∗=f∈A:zf′z/fz−1<1−β,S∗β,−β≔S∗β=f∈A:zf′z/fz−1<βzf′z/fz+1.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 Rezf′z/gz>β. In particular case, if f∈A and satisfies the condition Ref′z>β for all z in U, then fz is a close-to-convex of order β.Let1F1 denote the Kummer confluent hypergeometric function. The regular Coulomb wave function is defined as
(4)FL,ηz=zL+1e−izCLη1F1L+1−iη,2L+2;2iz=CLη∑n≥0aL,nzn+L+1,L,η∈ℂ,z∈ℂ,where
(5)CLη=2Le−πη/2ΓL+1+iηΓ2L+2=2L2L+1!2π∏k=0Lk2+η2ηe2πη−1,ifη≠0,2LL!2L+1!,ifη=0,aL,0=1,aL,1=ηL+1,aL,n=2ηaL,n−1−aL,n−2nn+2L+1,n∈2,3,⋯,which is the solution of following differential equation:
(6)z2ω′′z+z2−2ηz−LL+1ωz=0.In this paper, we focus on the following normalized form:(7)gL,ηz=CLη−1z−LFL,ηz.The functiongL,ηz satisfies the following homogenous second-order differential equation:
(8)z2gL,η′′z+2LzgL,η′z+z2−2ηz−2LgL,ηz=0.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 [2–5, 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.Lemma 1 (see [15, 16]).
LetΩ⊂ℂ, and Ψ:ℂ2×U⟶ℂ satisfy
(9)Ψiρ,σ;z∉Ω,whenever z∈U, ρ real, σ≤−1+ρ2/2. If p is analytic in U with p0=1, and Ψ=pz,zp′z;z∈Ω for z∈U, then Repz>0 in U. In the case Ψ:ℂ3×U⟶ℂ, then the condition in Lemma 1 generalized to
(10)Ψiρ,σ,u+iv;z∉Ω,
ρ real, σ+μ≤0, and σ≤−1+ρ2/2.Lemma 2 (see [19]).
Letf∈A. If
(11)Refzz>0,then
(12)Ref′z>0,for z<2−1.


## 2. Inclusion of Generalized Coulomb Wave Function in the Janowski Class

Our first result is related with Janowski starlikeness of normalized Coulomb wave function.Theorem 3.
Let−1≤B<A≤1 and L,η∈ℂ. Suppose that
(13)Re2L−1≤1+A4−ImL21+A2+A−21+η+Lfor−1=B<A≤1,or, for −1<B<A≤1,
(14)1+2η+2L1+B1+A−1+A1+B−A−B1+A1+B<Re2L−1,A−B21−B−21−B−1+2η+2L1−B2>Re2L−1,(15)2ImL1−AB2≤1+1+A1+BA−BRe2L−1+1+A2A−B−1+B2A−B1+2η+2L×21−BA−BRe2L−1−1+4A−B+1−B2A−B1+2η+2L.
If1+BzgL,η′z≠1+AgL,ηz,0≠gL,ηz and 0≠gL,η′z, then gL,η∈S∗A,B.Proof.
Define an analytic functionp:U⟶ℂ by
(16)pz≔1−AgL,ηz−1−BzgL,η′z1+BzgL,η′z−1+AgL,ηz,p0=1.
Then,(17)zgL,η′zgL,ηz=1−A+1+Apz1−B+1+Bpz,(18)zgL,η′′zgL,η′z=2A−Bzp′z1−A+1+Apz1−B+1+Bpz−1+zgL,η′zgL,ηz.
A rearrangement of (18) gives.
(19)zgL,η′′zgL,η′zzgL,η′zgL,ηz=2A−Bzp′z1−B+1+Bpz2−1−A+1+Apz1−B+1+Bpz+1−A+1+Apz21−B+1+Bpz2.
Now, define a functionqL,η:U⟶ℂ by
(20)qL,ηz=zgL,η′zgL,ηz.
This functionqL,η is analytic in U and qL,η0=1. Suppose that z≠0. We know that gL,ηz≠0. This function satisfies the following equation:
(21)z2gL,η′′zgL,ηz+2LzgL,η′zgL,ηz+z2−2ηz−2L=0.which yields
(22)zgL,η′′zgL,η′zzgL,η′zgL,ηz+2LzgL,η′zgL,ηz+z2−2ηz−2L=0.
Substituting (17) and (19) in (22), we get
(23)2A−Bzp′z1−B+1+Bpz2+1−A+1+Apz21−B+1+Bpz2+2L−11−A+1+Apz1−B+1+Bpz+z2−2ηz−2L=0,
or equivalently(24)zp′z+2L−12A−B1−A+1+Apz1−B+1+Bpz+1−A+1+Apz22A−B+z2−2ηz−2L2A−B1−B+1+Bpz2=0.
Now setting(25)Ψpz,zp′z;z≔zp′z+2L−12A−B1−A+1+Apz1−B+1+Bpz+1−A+1+Apz22A−B+z2−2ηz−2L2A−B1−B+1+Bpz2.
Then, forρ∈ℝand σ=−1+ρ2/2, we get
(26)ReΨiρ,σ;z=σ+Re2L−12A−B1−A+1+Aiρ1−B+1+Biρ+Re1−A+1+Aiρ22A−B+Rez2−2ηz−2L2A−B1−B+1+Biρ2≤−1+ρ22+Re2L−12A−B1−A1−B−1+A1+Bρ2−2ImLA−B1−ABρ+1−A2−1+A2ρ22A−B+1+2η+2L2A−B1−B+1+Biρ2=−ρ212+1+A1+B2A−BRe2L−1+1+A22A−B−1+B22A−B1+2η+2L−2ImLA−B1−ABρ+1−A1−B2A−BRe2L−1−12+1−A22A−B+1−B22A−B1+2η+2L≤−ρ212+1+A1+B2A−BRe2L−1+1+A22A−B−1+B22A−B1+2η+2L−2ImLA−B1−ABρ+1−BA−BRe2L−1−12+2A−B+1−B22A−B1+2η+2L≔Qρ.
To get the contradiction, we have to showQρ≤0 for ρ∈ℝ. We split the proof into two cases. First, consider the case B=−1<A≤1. Then, the function Q becomes
(27)Qρ=−ρ22+A2−2ImLρ+21+ARe2L−1−12+21+A+21+A1+2η+2L,
that achieve its maximum atρ0=−2ImL/2+A, and
(28)Qρ0=2ImL22+A+21+ARe2L−1−12+21+A+21+A1+2η+2L,which is nonpositive if and only if
(29)Re2L−1≤1+A4−ImL21+A2+A−21+η+L.
Now, consider the case−1<B<A≤1. Rewriting Q in the form
(30)Qρ=−Pρ2+Rρ−S=−Pρ−R2P2+4PS−R24P2,where
(31)P=12+1+A1+B2A−BRe2L−1+1+A22A−B−1+B22A−B1+2η+2L,R=−2ImLA−B1−AB,S=−1−BA−BRe2L−1−12+2A−B+1−B22A−B1+2η+2L.
The inequalityQρ≤0 holds for any real ρ, if P>0,S>0 and R2≤4PS or
(32)1+2η+2L1+B1+A−1+A1+B−A−B1+A1+B<Re2L−1,A−B21−B−21−B−1+2η+2L1−B2>Re2L−1,and
(33)2ImL1−AB2≤1+1+A1+BA−BRe2L−1+1+A2A−B−1+B2A−B1+2η+2L×21−BA−BRe2L−1−1+4A−B+1−B2A−B1+2η+2L,≤1+1+A1+BA−BRe2L−1+1+A2A−B−1+B2A−B1+2η+2L×21−BA−BRe2L−1−1+4A−B+1−B2A−B1+2η+2L,
that holds by hypothesis (14) and (15). Thus, in both cases, the function Ψ satisfies the hypothesis of lemma (8) and hence Repz>0, or
(34)1−AgL,ηz−1−BzgL,η′z1+BzgL,η′z−1+AgL,ηz≺1+z1−z.
By definition of subordination, there exist a mapω in Uwith ω0=0, and
(35)1−AgL,ηz−1−BzgL,η′z1+BzgL,η′z−1+AgL,ηz=1+ωz1−ωz,which yields
(36)zgL,η′zgL,ηz=1+Aωz1+Bωz.
Hence,(37)zgL,η′zgL,ηz≺1+Az1+Bz.If takeA=1−2β and B=−1 for 0≤β<1 in Theorem 3, we obtain following result.Corollary 4.
Let0≤β<1 andL,η∈ℂ. If
(38)ReL≤−1+β4−ImL21−β3−2β−η+L,
then the normalized Coulomb wave functiongL,ηz is starlike of order β.Theorem 5.
Let−1≤B<A≤1 and L,η∈ℂ satisfy
(39)Re2L+1≥1+2η1+A21+A2+1−A,−1=B<A≤3−22,1+2η1+A2AandRe2L+1≤1+2η1+A1−AA≠1,B=−1,A>3−22,1+2η1+A1−B2A−B1+B−1−B1+B,−1<B≤0,1+2η1+A1+BA−B−1−B1+B,B≥0.
If1+BsL,η≠1+A, then gL,ηz/z∈PA,B.Proof.
Define a functionsL,η:U⟶ℂ by
(40)sL,ηz=gL,ηzz.
The functionsL,η is analytic in U and sL,η0=1. Suppose that z≠0.. This function satisfies the following equation:
(41)z2sL,η′′z+2L+1zsL,η′z+z2−2ηzsL,ηz=0.
Define the analytic functionp::U⟶ℂ by
(42)pz=−1−A−1−BsL,ηz1+A−1+BsL,ηz.
Then, simple computation yields(43)sL,ηz=1−A+1+Apz1−B+1+Bpz,(44)sL,η′z=2A−Bp′z1−B+1+Bpz2,(45)sL,η′′z=2A−B1−B+1+Bpzp′′z−41+BA−Bp′2z1−B+1+Bpz3.
Thus, using (43)–(45), the differential equation (41) can be rewritten as
(46)z2p′′z−2z21+Bp′2z1−B+1+Bpz+2L+1zp′z+1−B+1+Bpz1−A+1+Apz2A−Bz2−2ηz=0.
AssumeΩ=0 and define Ψr,s,t;z by
(47)Ψr,s,t;z≔t−21+B1−B+1+Brs2+2L+1s+1−B+1+Br1−A+1+Ar2A−Bz2−2ηz.
It follows from (47) that Ψpz,zp′z,z2p′′z;z∈Ω. To ensure Repz>0 for z∈U, from Lemma 1, it is enough to establish ReΨiρ,σ,μ+iν;z≤0 in U for any real ρ, σ≤−1+ρ2/2,, and σ+μ≤0. Let z=x+iy∈U.. A computation yields
(48)ReΨiρ,σ,μ+iv;z=μ−21+B1−Bσ21−B2+1+B2ρ2+Re2L+1σ+1−B+1+Biρ1−A+1+Aiρ2A−Bz2−2ηz.
Sinceσ≤−1+ρ2/2. Thus,
(49)ReΨiρ,σ,μ+iv;z≤−Re2L+11+ρ22−1−B21+ρ2221−B2+1+B2ρ2+1−B+1+Biρ1−A+1+Aiρ2A−Bz2−2ηz≤−Re2L+11+ρ22−1−B21+ρ2221−B2+1+B2ρ2+1−B+1+Biρ1−A+1+Aiρ2A−B1+2η.
The proof will be divided into four cases. Consider firstB=−1,B<A≤3−22. The inequality (49) reduces to
(50)ReΨiρ,σ,μ+iv;z≤−Re2L+11+ρ22+1−A+1+Aiρ1+A1+2η≤−Re2L+11+ρ22+1+2η1+A1−A+1+Aρ=−Re2L+12ρ2+1+2ηρ+1−A1+A1+2η−Re2L+12=−Re2L+12ρ−1+2ηRe2L+12+1+2η22Re2L+1+1−A1+A1+2η−Re2L+12≕Gρ.
A quadratic functionG takes nonpositive values for any ρ, if
(51)1+2η22Re2L+1+1−A1+A1+2η−Re2L+12≤0.
Last inequality can be rewritten as(52)−Re2L+12+21−A1+A1+2ηRe2L+1+1+2η2≤0,that holds, if
(53)Re2L+1≥1+2η1+A1−A2+1+A2−1−A,which reduces to the assumption. Therefore, the assertion follows.
In second case, we considerB=−1,A>3−22. According to (49), we have
(54)ReΨiρ,σ,μ+iv;z≤−Re2L+11+ρ22+1−A+1+Aiρ1+A1+2η=−Re2L+11+ρ22+1+2η1+A1−A2+1+A2ρ2≕Hρ.
We note that the functionH is even with respect to ρ, and
(55)H0=−Re2L+12+1−A1+A1+2η,that satisfies H0≤0, if
(56)Re2L+1≥21−A1+A1+2η.
Moreover,limρ⟶∞Hρ=−∞, and
(57)H′ρ=−Re2L+1ρ+1+Aρ1+2η1−A2+1+A2ρ2,with H′ρ=0 if and only if ρ=0 or
(58)ρ02=1+2η2Re22L+1−1−A21+A2.
We observe thatρ02≥0 by the inequality
(59)1+2η2Re22L+1≥1−A21+A2,(60)Re2L+1≤1+A1−A1+2η.
Additionally,(61)H′′ρ=−Re2L+1+1−A21+A2Re32L+11+2η2≤0,in view of (60). Hence, Hρ0=maxHρ, and
(62)Hρ0=−Re2L+121−1−A1+A2+1+2η22Re2L+1≤0,that holds if
(63)Re2L+1≥1+A2A1+2η.
Since,(64)1+A1−A1+2η≥1+A2A1+2η≥1−A1+A1+2η,holds for 3−22≤A≤1, then the condition (56), (60), and (63) reduce to the assumption (39). Therefore, the assertion follows. Let now −1<B≤0,A>B. By the fact 1−A/1+A<1−B/1+B, we get
(65)1−A+1+Aiρ1−B+1+Biρ=1+A1+B1−B1+B2+ρ21−A1+A2+ρ2≤1+A1+B1−B1+B2+ρ2.
Also, forB≤0, we have 1+B/1−B≤1; therefore,
(66)1+ρ21−B2+1+B2ρ2=11−B21+ρ21+1+B/1−B2ρ2≥11−B2,for any real ρ. Thus,
(67)ReΨiρ,σ,μ+iv;z≤−Re2L+11+ρ22−1+B1+ρ221−B+1+2η2A−B1+A1+B1−B1+B2+ρ2=ρ2−Re2L+12−1+B21−B+1+2η2A−B1+A1+B−Re2L+12−1+B21−B+1+2η2A−B1+B1+A1−B2..
Since forB≤0,,
(68)−Re2L+12−1+B21−B+1+2η2A−B1+A1+B≤−Re2L+12−1+B21−B+1+2η2A−B1+B1+A1−B2,
and the last expression is nonpositive in view of (39); then, the assertion follows. Finally, consider 0≤B<A≤1. In this case β=1+B/1−B≤1. Hence, setting t=β2+ρ2 with t≥β2 and using (65), we get from (49)
(69)ReΨiρ,σ,μ+iv;z≤−Re2L+121−β2+t−β1−β2+t22t+1+2η2A−B1+A1+Bt=t−Re2L+12−β2+1+2η2A−B1+A1+B−Re2L+121−β2−β1−β222t−β1−β22,
that is nonpositive due to inequality(70)Re2L+1≥1+2ηA−B1+A1+B−1−B1+B,that is equivalent to the assumption (39). Evidently, Ψ satisfies the hypothesis of Lemma 1, and thus, Repz>0, that is
(71)−1−A−1−BsL,ηz1+A−1+BsL,ηz≺1+z1−z.
Hence, there exists an analytic self-mapω of U with ω0=0 such that
(72)−1−A−1−BsL,ηz1+A−1+BsL,ηz=1+ωz1−ωz,which implies that
(73)gL,ηzz≺1+Az1+Bz.If we takeA=1−2β and B=−1 for 0≤β<1 in Theorem 5, we obtain following result.Corollary 6.
Let0≤β<1 and L,η∈ℂ. If
(74)1+2η1−β21−2β−12≤ReL≤1+2η1−β2β−12,0≤β<2−1,(75)ReL≥12+1+2η21−β1−2β+2β2+β,2−1≤β<1,
thenRegL,ηz/z>β, that is, z+∫0zgL,ηt/tdt is close-to-convex of order β.Applying Corollary6 for β=0 and Lemma 2, the following result for close-to-convexity of gL,ηz immediately follows.Corollary 7.
LetL,η∈ℂ. If ReL≥η, then gL,ηz is close-to-convex (univalent) for z<2−1.



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