# Starlikeness of Normalized Bessel Functions with Symmetric Points

Authors: Maryam Nazir; Syed Zakar Hussain Bukhari; Imtiaz Ahmad; Muhammad Ashfaq; Malik Ali Raza
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Bessel functions are related with the known Bessel differential equation. In this paper, we determine the radius of starlikeness for starlike functions with symmetric points involving Bessel functions of the first kind for some kinds of normalized conditions. Our prime tool in these investigations is the Mittag-Leffler representation of Bessel functions of the first kind.


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## Body

## 1. Introduction and Definitions

LetEz0,r=z∈ℂ:z−z0<r and E0,1⊂Ez0,r denote the interior of the unit circle with center at origin. Suppose that A represent functions f in E0,1:(1)fz=z+a2z2+⋯.Obviously,f0=0 along with f′0=1. The subclass S⊂A only contains univalent (one-to-one) functions and S∗⊂S represent the set of starlike functions. A function f for which fE0,1 is star-shaped is starlike if Rezf′z/fz>0. Also, f∈Ss if(2)Re2zf′zfz−f−z>0,z∈E0,1,and f∈Ssη if(3)Re2zf′zfz−f−z>η,0≤η<1,z∈E0,1.Let(4)r∗f=supr>0:Re2zf′zfz−f−z>0,z∈E0,r,rη∗f=supr>0:Re2zf′zfz−f−z>η,z∈E0,rbe the radii of the classes defined above. We note that r∗ is the maximum value of the radius such that fE0,r∗f∈S∗ and rη∗ is the maximum value of the radius such that fE0,rη∗f∈S∗ with symmetric points. Consider the following representation of the function bμ as in [1], which satisfies the well-known Bessel differential equation:(5)bμz=∑j≥0−1jj!Γj+μ+1z22j+μ=∑j≥0j!−1jj+μ!z22j+μ,where z,μ∈ℂ such that μ≠−1,−2,−3,⋯. Observe that bμz∉A. Thus, we consider the following normalizations:(6)fμz=2μΓμ+1bμz1/μ,μ≠0,(7)kμz=2μΓμ+1z1−μbμz,(8)ℓμz=2μΓμ+1z1−μ/2bμz.Clearly, the functionfμ,kμ,ℓμ∈A. We see that(9)fμz=exp1μlog2μΓμ+1bμz.The geometric behavior and properties of the functionsfμ, kμ, and ℓμ were studied by Brown, Kreyszig, Robertson, and many others (for detail, see [2–4] and also the references therein). The related problems were also studied in [3, 5–8] with references therein. We study the radius problems for the functions fμ, kμ, and ℓμ starlike with symmetric points. Mittag-Leffler expansion for Bessel functions is used as a prime tool along with the conclusion that the specific positive roots of the Dini functions are always smaller than the related zeros bμz, for reference, see [9].


## 2. Preliminaries

Lemma 1.
Letf:E0,1⟶ℂ be a transcendental function having the following expansion:(10)fz=z∏j≥11−zzj,where zj:zj>1 have the same argument. For a univalent function f in E0,1, we have(11)∑j≥11zj−1≤1.This result holds if and only iff∈S∗, and each of its derivatives is close to convex in the open unit disk E0,1. Furthermore, for zj′, the zeroes of the derivative of f, f, and f′ are univalent in E0,1 and for E0,1,fE0,1 is a convex-shaped if and only if(12)∑j≥11zj′−1≤1.Lemma 2.
The function(13)ℓμz=2μΓμ+1z1−μ/2bμz∈S∗ηand each of its derivative is close to convex in E0,1 if and only if μ>μ0η, where μ0η≃0.5623⋯ is a unique zero of ℓμ1=0 on −1,∞.The proof of Lemma1 and Lemma 2 is found in [10].Lemma 3.
The function(14)fμz=2μΓμ+1bμz1/μ∈S∗ηin E0,1 if and only if μ>μ1η, where 0<μ1η<∞ is the unique solution of(15)1−ημbμ1=bμ+11.
In particular,fμ∈S∗ in E0,1 if and only if μ>μ10, where μ10≃0.3908⋯ is a unique zero of(16)μbμ1=bμ+11.Lemma 4.
The function(17)kμz=2μΓμ+1bμzz1−μ∈S∗η,0≤η<1,in E0,1 if and only if μ>μ2η, where μ2η is the unique zero of(18)1−ηbμ1=bμ+11,lies in μ~,∞, where μ~≃−0.7745⋯ is the unique root of bμ,1=1 and bμ,1 is the first positive zero of bμ. In particular, kμ∈S∗ in E0,1 if and only if μ>μ20, where μ20≃−0.3397⋯ is a unique zero of(19)bμ1=bμ+11.The proof of Lemma3 and Lemma 4 can be seen in [11].Lemma 5.
Ifz∈ℂ and η∈ℝ:η>z, then(20)zη−z≥Rezη−z.For the detail of the above Lemma5, we refer to [3].


## 3. Main Results

Theorem 6.
Let1>η≥0, and μ∈−1,0. Then, rη∗Iμ, is a unique positive zero of(21)zIμ′z−ημIμz=0,where Iμz=i−μbμiz. Moreover, if μ>0, then rη∗bμ is the least positive zero of(22)zbμ′z−ημbμz=0.Proof.
Using Lemma3, we see that the function fμz=2μΓμ+1bμz1/μ∈S∗η in E0,1 with respect to z iff μ>μ1η, where μ1η is a unique zero of(23)1−ημbμ1=bμ+11,lies in 0,∞. Suppose bμ,j is the jth positive root of bμz. By using infinite product representation,(24)bμz=1Γ1+μz2μ∏j≥11−z2bμ,j2.Also, as given in [12], we see that fμ has the following form:(25)fμz=z∏j≥11−zbμ,j2=∑j≥0−1jΓ1+μj!Γμ+j+1zμ+2j1/μ=z−14μμ+1z3+⋯,μ≠0.
From Lemma3, we see that for μ>μ∗≃−0.7745⋯ the unique value of the root of fμ1=0 or bμ,1=1, we have bμ,1>1, and bμ,j>1,j=1,2,⋯. The above result is immediate, if bμ,1 is increasing on −1,∞. Using (24) and (25), we can write(26)−bμz=bμ−z,−bμ′z=bμ′−z.
Also, from (6), we have(27)−fμz=fμ−z,fμ′−z=−1fμ′z,which in the context of zbμ′z−μbμz=−zbμ+1z is equivalent to the Mittag-Leffler representation:(28)1bμzbμ+1z=∑j≥12zbμ,j2−z2.
Consequently,(29)bμ+1−zbμ−z=−bμ+1zbμz.
In view of (6), (25), and (27), we can write(30)zfμ′−zfμ−z=−1+1μ∑j≥12z2bμ,j2−z2=−=−1μzbμ′zbμz.
Using Lemma1, we find that for j∈ℕ,μ=−1, and z∈E0,bμ,1, the following inequality(31)z2bμ,j2−z2≥Rez2bμ,j2−z2implies that(32)Rezfμ′−zfμ−z≤−1+1μ∑j≥12z2bμ,j2−z2=−zfμ′zfμz.
Whenz<bμ,1, we observe that(33)Rezfμ′−zfμ−z≤−1+1μ∑j≥12r2bμ,j2−r2≤−1+1μ∑j≥12bμ,j2−1=−fμ′1fμ1.
As in [12], we see that bμ,j′>0 on 0,∞ for a fixed j∈N. Thus, fμ′1/fμ1 is increasing on 0,∞, and −fμ′1/fμ1 is decreasing on 0,∞. Also,(34)−fμ′1fμ1<−η⇔μ<μ1η,where μ1η is the unique zero of(35)fμ′1=ηfμ1orημbμ1=bμ′1or1−ημbμ1=bμ+11.
We also note that(36)zfμ′−zfμ−z=1−bμ+1zμbμz,when μ∈−1,0. For −1<μ<−∞, the Dini function zbμ′z+ηbμz has real roots except a pair of complex conjugate roots (for detail, see [1]). Thus,(37)fμ−z∈S∗η,η∈−1,0,in E0,1 if and only if μ<μ1η. Considering (5), (30), (33), and (36), we have(38)2zfμ′zfμz−fμ−z=2zfμzbμ′zμfμz+fμzbμz=1μzbμ′zbμz=1−1μ∑j≥12z2bμ,j2−z2.
Also, from (38), it is obvious that(39)Re2zfμ′zfμz−fμ−z≥1−1μ∑j≥12z2bμ,j2−z2=zfμ′zfμz=Rezfμ′zfμz.
As in [1], for 1>η≥0 and μ∈−1,0, rη∗Iμ is the unique value of positive zero of zIμ′z−ημIμz=0. Moreover, if μ>0, then we have rη∗bμ which is the least positive zero of zbμ′z−ημbμz=0.Theorem 7.
Ifμ>−1, then rη∗kμ>0 is the smallest zero of(40)zbμ′z+1−η−μbμz=0,where kμz=2μΓμ+1bμzz1−μ.Proof.
By Lemma4, the function(41)kμz=2μΓμ+1bμz1−μ∈S∗η,for η∈−1,0in the open unit disk E0,1 if and only if μ<μ2η, where μ2η is the unique value of the zero of the following equation:(42)1−ηbμ1=bμ+11.
Suppose thatbμ,j is the jth positive zero of bμz given by (24) and (25). Consider the normalization (7) such that(43)kμ−z=−1kμz.
We write(44)zkμ′−zkμ−z=−zkμ′zkμz=−1+μ−∂bμz=−1+zbμ+1zbμz.
Forμ>−1 and r=z<bμ,1, we see that(45)Rezkμ′−zkμ−z≤−zkμ′rkμr=−1+∑j≥12z2bμ,j2−z2=−kμ′1kμ1.
For detail, we refer to [12]. Since the function bμ,j′>0 on −1,∞ for fixed j∈ℕ, thus kμ′1/kμ1 is increasing on μ~,∞, and −kμ′1/kμ1 is decreasing on μ~,∞ and −kμ′1/kμ1<−η if and only if μ<μ2η, where μ2η is the unique value of the root of(46)kμ′1=ηkμ1or1−μ−ηbμ1+bμ′1=0or1−ηbμ1=bμ+11.
Thus,(47)zbμ′−zbμ−z=−1+Re∑j≥12z2bμ,j2−z2≤−1+∑j≥12z2bμ,j2−z2=−zkμ′zkμz=−Rezkμ′zkμz,and equality holds for z=r. The above inequality implies that the function kμ−z∈S∗η,η∈−1,0, in E0,1 if and only if μ<μ2η. Considering normalization in (7), we can write(48)kμ−z=−1kμz2zkμ′zkμz−kμ−z=zbμ′zbμz+1−μ=1−∑j≥12z2bμ,j2−z2.
From (48), it is known that(49)Re2zkμ′zkμz−kμ−z=1−Re∑j≥12z2bμ,j2−z2≥1−∑j≥12z2bμ,j2−z2=zkμ′zkμz=Rezkμ′zkμz.
For1>η≥0 and μ>−1, we see that rη∗kμ is the least positive zero of the following equation:(50)zbμ′z+1−η−μbμz=0,where kμz=2μΓμ+1bμzz1−μ.Theorem 8.
Ifμ>−1 and 1>η≥0, then rη∗ℓμ is the least positive zero of the differential equation zbμ′z+2−2η−μbμz=0, where ℓμz=2μΓμ+1bμzz1−μ/2.Proof.
Assume thatbμ,j is the jth positive zero of bμz given by (24) and (25). Considering the normalization given in (7), we write(51)ℓμ−z=2μΓμ+1bμ−z−z1−μ/2.
Since(52)ℓμz=2μΓμ+1bμzz1−μ/2=∑j≥0−1jΓ1+μ4jj!Γμ+j+1z1+j=z∏j≥11−zbμ,j2, so we can write(53)Rezℓμ′zℓμz=1−μ2+Re12zzbμ′zbμz=1−Re∑j≥1zbμ,j2−z.
This result shows that(54)Rezℓμ′−zℓμ−z=−1+μ2−12−z−zbμ′−zbμ−z=−1+∑j≥1−zbμ,j2+z,or(55)Rezℓμ′−zℓμ−z=−1−Re∑j≥1zbμ,j2+z≤−1+∑j≥1zbμ,j2−z=−zℓμ′zℓμz=−Rezℓμ′zℓμz.
The equality holds forr=z=z. The principle of minimum value for harmonic functions along with (7) shows that(56)Rezℓμ′zℓμz≥1−∑j≥1zbμ,j2−zis valid if and only if z<bμ,1, and bμ,1 is the minimum positive root of the equation(57)rbμ′r+2−μbμr=0.
Thus, we have(58)ℓμ′−1ℓμ−1=−1+∑j≥111−−1/bμ,j21bμ,j2=−1−∑j≥11bμ,j2+1≤0,(59)ℓμ′−zℓμ−z=−1z−∑j≥11bμ,j2+zor∂∂μℓμ′−zℓμ−z=−∑j≥12bμ,j∂/∂μbμ,jz+bμ,j22,or(60)∂∂μℓμ′−1ℓμ−1=−∑j≥12bμ,j∂/∂μbμ,j1+bμ,j22≤0,since bμ,j′>0 on −1,∞ for a fixed j∈ℕ. Thus, by using (58) and (60), we see that ℓμ satisfies (7) and by applying Lemma 1 and Lemma 2, we obtain that ℓμ−z∈S∗ and decreasing on −1,∞ and by considering (8), we can write(61)zℓμ′z=2μΓμ+1zbμ′z2zz−1+μ/2+1−μ2bμzz−1+μ/2=∑j≥0−1jj+1Γ1+μ4jj!Γμ+j+1z−j−1.
We also write(62)ℓμz−ℓμ−z=∑j≥01−−11+j−1jΓ1+μ4jj!Γj+μ+1z1+j.
We can write(63)2zℓμ′zℓμz−ℓμ−z=∑j≥0j+1−1jΓ1+μzj+1/4jj!Γμ+j+1∑j≥01−−11+j−1jΓ1+μzj+1/4jj!Γ1+j+μ=21+−iμbμiz/bμzzℓμ′zℓμz.
Since(64)bμ−z=∑j=0∞−1j−z/2μ+2jΓ1+jΓ1+μ+j=∑j=0∞iμ−12jΓj+1Γ1+μ+jz2μ+2j,so we have(65)1+bμizbμz−iμ=∑j=0∞−1j1+−1j/Γj+1Γ1+μ+jz/2μ+2j∑j=0∞−1j/Γ1+jΓμ+j+1z/2μ+2j.
From (63) along with (65), we see that(66)Re2zℓμ′zℓμz−ℓμ−z=Rezℓμ′zℓμz1+bμizbμz−iμ−1=Rezℓμ′zℓμz1−bμiz−iμbμz+⋯=Rezℓμ′zℓμz∑j=0∞−1j/Γj+1Γ1+μ+jz/2μ+2j1+−1j∑j=0∞−1j/Γj+1Γ1+μ+jz/2μ+2j≥1−∑j≥12zbμ,j2−z=zℓμ′zℓμz.
As in [11], we observe that(67)Re2zℓμ′zℓμz−ℓμ−z≥Rezℓμ′zℓμz≥1−∑j≥12zbμ,j2−z=zℓμ′zℓμz.
For1>η≥0 and μ>−1, rη∗ℓμ is the least positive zero of(68)zbμ′z+2−2η−μbμz=0,where ℓμz=2μΓμ+1bμzz1−μ/2.


## 4. Conclusion

The class of Bessel functions is originated as a solution of the well-known Bessel differential equation. We studied the radius problems of starlike functions with symmetric points involving Bessel functions under some kind of normalized conditions. We used the Mittag-Leffler representation of Bessel functions and derived our main results.



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