# On Bessel Functions Related with Certain Classes of Analytic Functions with respect to Symmetrical Points

Authors: Afis Saliu; Khalida Inayat Noor; Saqib Hussain; Maslina Darus
Journal: Journal of Mathematics
 (2021)

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## Abstract

In the present investigation, subclasses of analytic functions with respect to symmetrical points which are defined by the generalized Bessel functions of the first kind of orderμ are introduced. Furthermore, some alluring geometric properties of these classes, which include inclusion property, integral-preserving properties, coefficients, and distortion results are studied. Moreover, some consequences of our results are also given.


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## Body

## 1. Introduction

Geometric function theory (GFT) is the area of complex analysis which deals with the geometric characterization of analytic functions, established around the turn of the twentieth century[1]. It is a known fact that the study of special functions plays a significant role in GFT. One reason is that solutions of extremal problems can be frequently written in terms of special function. Another reason is that some important conformal mappings are given by special function. For example, the conformal mapping of an annulus onto the complement of two closed segments on the real axis and the conformal mapping of a square onto a rectangle are expressed by elliptic functions (see [2]). In recent times, the solution of Bieberbach conjecture by de Branges is obtained with the help of special functions [3].Bessel function is one of the most significant special functions. It is therefore important for solving many problems in engineering, physics, and mathematics (see [4, 5]). For instance, it is used for velocity and stress derivation in the rotational flow of Burge’s fluid flowing through an unbounded round channel [6].In recent times, many researchers paid their attention on establishing various conditions under which a Bessel function has some certain geometric properties such as close-to-convexity (univalency), starlikeness, and convexity in frame of a unit discU (see [7–11]).The objective of this manuscript is twofold. Firstly, Bessel functions of the first kind of orderμ is used to introduce new generalized starlike and convex functions with respect to symmetrical points, which was first initiated and studied by Sakaguchi [12] and Das and Sign [13]. Moreover, we examine some interesting geometric properties of these classes, which include inclusion property, integral-preserving properties, coefficients, and distortion results.


## 2. Materials and Methods

Now, we give some basic preliminaries and definitions that play the integral part in obtaining our main results.Considerb,c,μ∈ℂ  (the set of complex numbers) and the second-order linear homogenous differential equation(1)z2w″z+bzw′z+cz2−μ2+1−bμwz=0,which is a natural extension of Bessel’s equation. The solution wz (see [14]) of (1) has a series representation:(2)wz=∑n=0∞−cnn!Γμ+n+b+1/2z22n+μ,z∈ℂ.Differential equation (2) allows the investigation of Bessel function of the first kind of order μ [15, 16] (the case b = c = 1), modified Bessel function [15, 16] (the case b = 1, c = −1), and the spherical Bessel Function [16] (the case b = 2, c = 1). Using the well-known Pochhammer symbol an with a≠0,−1,−2,…, we consider the function φμ,b,cz defined by the transformation(3)φμ,b,cz=2μΓμ+b+12z1−μwz=z+∑n=2∞−cn−14n−1μ+b+1/2n−1anzn,whereμ+b+12≠0,−1,−2,….LetA denote the class of normalized analytic functions fz in U given by the representation(4)fz=z+∑n=2∞anzn.Then, the convolution offz and gz=z+∑n=2∞bnzn denoted by fz∗gz is defined by(5)fz∗gz=z+∑n=2∞anbnzn,and we say fz is subordinate to gz (written as fz≺gz) if there exists a Schwarz function wz such that fz=gwz,z∈U.Letℬμ,bc:A⟶A be an operator defined by(6)ℬμ,bcfz=φμ,b,cz∗fz=∑n=1∞Anzn,whereAn=−cn−14n−1μ+b+1/2n−1n−1!an,a1=1.From (6), we have the identity relation(7)zℬμ+1,bcfz′=μ+pℬμ,bcfz−μ+p−1ℬμ+1,bcfz,where p=b−1/2. It is easy to observe from (7) that(8)zΦμ+1,bcfz′=μ+pΦμ,bcfz−μ+p−1Φμ+1,bcfz,where(9)Φμ,bcfz=12ℬμ,bcfz−ℬμ,bcf−z.Letϕz be a convex univalent function in U with ϕ0=1 and Reϕz>0 in U. Ma and Minda and Kim examined the classes Cϕ,S∗ϕ (see [17]), and Kϕ (see [18]) using the subordination techniques. In particular, for −1≤B<A≤1, C1+Az/1+Bz≡CA,B,S∗1+Az/1+Bz≡S∗A,B [19] and K1+Az/1+Bz≡KA,B [20] and S∗1+z/1−z≡S∗ and C1+z/1−z≡C [21].Definition 1.
Letf∈A. Then, f∈Sμ,b,csϕ if and only if(10)zℬμ,bcfz′Φμ,bcfz≺ϕz,z∈U,and f∈Cμ,b,csϕ if and only if(11)zℬμ,bcfz′′Φμ,bcfz′≺ϕz,z∈U.
We note thatf∈Cμ,b,csϕ⟺zf′∈Sμ,b,csϕ. If ϕz=1+Az/1+Bz,−1≤B<A≤1, we set Sμ,b,cs1+Az/1+Bz=Sμ,b,csA,B and Cμ,b,cs1+Az/1+Bz=Cμ,b,csA,B. It is worthy of note that if −cn−1/4n−1μ+b+1/2n−1n−1!=1 in (6) and ϕz=1+z/1−z, then the classes Sμ,b,csϕ and Cμ,b,cϕ reduce to the classes Ss∗ and Cs, consisting of functions which are starlike and convex with respect to symmetrical points [12, 13, 22–24].
The following lemmas are the key tools to prove our main results.Lemma 1.
(see [19, 25]). If pz≺1+Az/1+Bz,−1≤B<A≤1, then(12)12π∫02πpz2dθ≤1+B−A2B2B2r21−B2r2,B≠0,1+A2r2,B=0,(13)1−Ar1−Br≤pz≤1+Ar1+Br.Lemma 2.
(see [26]). Let gz be an univalent function in U. Then, there exists z1 with z1=r such that for all z∈U,(14)z−z1gz≤2r21−r2.Lemma 3.
(see [21]). Let ψz be convex in U with ψ0=1. Suppose also that λz is analytic in U with Reλz≥0z∈U. If pz is analytic in U with p (0) = 1, then(15)pz+λzzp′z≺ψzinU,which implies that(16)pz≺ψzinU.Lemma 4.
(see [21, 27]). Let ψz be convex in U with Reβψz+γ>0. If pz is analytic in U with p0=ψ0, then(17)pz+zp′zβpz+γ≺ψz⇒pz≺ψzinU.


## 3. Results and Discussion

Theorem 1.
Iff∈Sμ,b,csϕ, then Φμ,bcfz∈S∗ϕ.Proof.
Letf∈Sμ,b,csϕ. Then,(18)zℬμ,bcfz′Φμ,bcfz=pz,pz≺ϕz.
Replacingz with −z in (18) and using the fact that Φμ,bcfz is an odd function, we have(19)zℬμ,bcf−z′Φμ,bcfz=p−z,p−z≺ϕz,which combined with (18) gives(20)zΦμ,bcfz′Φμ,bcfz=pz,pz+p−z.
By subordination property, we have thatΦμ,bcfz∈S∗ϕ.Corollary 1.
The functionℬμ,bcfz belongs to Kϕ and hence is univalent in U.Setting−cn−1/4n−1μ+b+1/2n−1n−1!=1 and choosing ϕz=1+z/1−z in Theorem 1, we are led to the result of Sakaguchi [12] contained in the following corollary.Corollary 2.
Every functionfz in Ss∗ is a close-to-convex function.Theorem 2.
SupposeReμ+b+1/2>1 and Φμ,bcfz∈S∗ϕ. Then,(21)Sμ,b,csϕ⊂Sμ+1,b,csϕ.Proof.
Consider(22)zℬμ,bcfz′Φμ,bcfz=hz.
From relation (7), (22) can be written as(23)μ+pℬμ,bcfz=hzΦμ+1,bcfz+μ+p−1ℬμ+1,bcfz.
Differentiating (23) and applying (8), we obtain(24)μ+pzℬμ,bcfz′=zh′zΦμ+1,bcfz+μ+phzΦμ,bcfz,i.e.,(25)μ+pzℬμ,bcfz′Φμ,bcfz=μ+phz+zh′zΦμ+1,bcfzΦμ,bcfz.
It follows from (8) that(26)Φμ+1,bcfzΦμ,bcfz=μ+pHz+μ+p−1,where(27)Hz=zΦμ+1,bcfz′Φμ+1,bcfz≺ϕz.
In view of (25) and (26), we obtain(28)zℬμ,bcfz′ℬμ,bcfz=hz+zh′zHz+μ+p−1.
SinceReμ+b+1/2>1 and Φμ+1,bcfz∈S∗ϕ, then ReHz+μ+p−1>0. Hence, by Lemma 3, hz≺ϕz, i.e., f∈Sμ,b,csϕ. This completes the proof.Corollary 3.
LetReμ+b+1/2>1 and Φμ+1,bcfz∈S∗A,B. Then,(29)Sμ,b,csA,B⊂Sμ+1,b,csA,B.Corollary 4.
Suppose that all the conditions of Theorem2 are satisfied. Then,(30)Cμ,b,csϕ⊂Cμ+1,b,csϕ.Proof By Theorem2.
we have that(31)f∈Cμ,b,csϕ⟺zf′∈Sμ,b,csϕ⟺zf′∈Sμ+1,b,csϕ⟺f∈Cμ+1,b,csϕ.Theorem 3.
LetJαfz be defined by the integral transformation(32)Jαfz=α+12zα∫0ztα−1ft−f−tdt,α=1,2,3,….and suppose f∈Sμ,b,csϕ. Then, ℬμ,bcJαfz∈S∗ϕ.Proof.
Let(33)zℬμ,bcJαfz′Φμ,bcJαfz=pz,where pz is analytic in U with p0=1. From (32) and applying the operator ℬμ,bc, we obtain(34)zℬμ,bcJαfz′−αℬμ,bcJαfz=α+1Φμ,bcfz,i.e.,(35)ℬμ,bcJαfzpz+α=α+1Φμ,bcfz.
Differentiating (35) logarithmically, we obtain(36)pz+zp′zpz+α=zΦμ,bcfz′Φμ,bcfz.and in view of Theorem 1, it follows that(37)pz+zp′zpz+α≺ϕz,which by Lemma 4 implies pz≺ϕz. Thus, ℬμ,bcJαfz∈S∗ϕ.Corollary 5.
Letf∈Cμ,c,bsϕ. Then, ℬμ,bcJαfz∈Cϕ.Proof.
(38)f∈Cμ,c,bsϕ⟺zf′zf∈Sμ,c,bsϕ⟺ℬμ,bcJzf′z∈S∗ϕby Theorem3⟺zℬμ,bcJαfz′∈S∗ϕ⟺ℬμ,bcJαfz∈Cϕ.Corollary 6.
Let−1≤B<A≤1 and f∈Sμ,c,bsA,B. Then, ℬμ,bcJαfz∈S∗A,B. Similarly, if f∈Cμ,c,bsA,B, then ℬμ,bcJαfz∈C∗A,B.Setting−cn−1/4n−1μ+b+1/2n−1n−1!=1 and choosing ϕz=1+z/1−z in Theorem 3, we are led to the results of Das and Sign [13] contained in the following corollaries.Corollary 7.
Letf∈Ss∗. Then, J0fz∈S∗.Corollary 8.
Letf∈Cs. Then, J0fz∈C.Theorem 4.
Iff∈Sμ,c,bsA,B, then for c≠0,(39)an≤D1A,B4n−1μ+b+1/2n−1n−1!n−A/2Bcn−1,B≠0,D1A4n−1μ+b+1/2n−1n−1!ncn−1,B=0.,where D1A,B is a constant that depends on A and B, while D1A only depends on A.Proof.
Sincef∈Sμ,c,bsA,B, then by Cauchy Theorem,(40)nAn≤12πrn∫02πΦμ,bcfzpzdθ,where An is given by (6) and pz≺1+Az/1+Bz,z∈U. From Theorem 1, we knew that for ϕz=1+Az/1+Bz, the function Φμ,bcfz is an odd starlike function of Janowski type. Thus,(41)Φμ,bcfz≺z1+Bz2B−A/2B,B≠0,zexpAz22,B=0.
By Cauchy–Schwarz inequality, subordination property, and Lemma1 for the case B≠0, (40) implies(42)nAn≤1rn12π∫02πΦμ,bcfz2dθ1/212π∫02πpz2dθ1/2,z=reiθ≤1rn12π∫02πr21+Br2A−B/2Bdθ1/21+B−A2r21−B2r21/2=rrn1+Br2A−B/2B1+B−A2r21−B2r21/2.
Observe that since−1≤B<1, we have(43)11+Br2≤11−r2,11−Br2<11−r2.
Thus,(44)nAn≤1+B−A2rn1−rB−A/2B+1/2.
Letr=1−1/nn⟶∞; we obtain from (6) and (44) that(45)an≤D1A,B4n−1μ+b+1/2n−1n−1!n−A/2Bcn−1,where D1A,B is given in Theorem 4.
For the caseB=0, we implement Lemma 1 and subordination property in (41) and follow the procedures for the case B≠0 to obtain(46)nAn≤expA2+11+A2,which completes the proof by using (6).Remark 1.
If we allow−cn−1/4n−1μ+b+1/2n−1=1 and choose A=1,B=−1 in Theorem 3, it follows that(47)an=O1n1/2,whereO1isaconstant.
SinceSs∗ is a subclass of the class of close-to-convex function (see [12]), it shows that our index of n is a nice one.Theorem 5.
Letf∈Sμ,b,csA,B. Then,(48)an+1|−|an|≤D2A,B4n−1μ+b+1/2n−1n−1!ncn−1,B≠0,D2A4n−1μ+b+1/2n−1n−1!cn−1,B=0,where D2A,B is a constant that depends on A and B, while D2A only depends on A.Proof.
Letz1 be a complex number with z1=r. Then, by Cauchy Theorem, we obtain(49)z1n+1An+1−nAn≤12πrn+1∫02πz1−zzℬμ,bcfz′dθ=12πrn+1∫02πz1−zΦμ,bcfzpzdθ,where pz≺1+Az/1+Bz,−1≤B<A≤1. Following the techniques used in Theorem 4 and using Lemma 2, we obtain(50)z1n+1An+1−nAn≤2rn−11−r1+B−A21−r1/2.
Takingr=1−1/nn⟶∞ and choosing z1=n−c/4n−1/μ+b+1/2n−1n−1!/n+1−c/4n/μ+b+1/2nn!, we have the result. The case B=0 also follows the same procedures.Theorem 6.
Iff∈Sμ,b,csA,B, then(51)2n−1−−1n2c2an24n−1μ+b+1/2n−1n−1!≤∑m=1n−1A1−−1m−2Bm2c2am24m−1μ+b+1/2m−1m−1!,B≠04A2c2am24m−1μ+b+1/2m−1m−1!,B=0.Proof.
Letf∈Sμ,b,csA,B. Then, there exists an analytic function wz=∑n=1∞wnzn with w0=0 and wz<1z∈U such that(52)zℬμ,bcfz′Φμ,bcfz=1+Awz1+Bwz,which is equivalent to(53)zℬμ,bcfz′−Φμ,bcfz=AΦμ,bcfz−Bzℬμ,bcfz′wz.
Thus,(54)∑n=1∞2n−1−−1nAnzn=∑n=1∞A1−−1n−2BnAnznwz,where we have used (6), (9), and (53). Comparing the coefficients of zn in (54), we obtain(55)2A−BA1wn−1−2BA2w2+⋯−A1−−1n−2Bn−1An−1w1=2n−1−−1nAn.
The coefficients combination on the right side of (55) depends only upon the coefficients combination 2A−BA1,−2BA2,2A−3BA3,…,A1−−1n−2Bn−1An−1 of the left side. Therefore, we can write (54) as(56)wz∑m=1n−1A1−−1m−2BmAmzm=∑m=1n2m−1−−1mAmzm+∑m=n+1∞cmzm,for some cm,n+1≤m<∞. Squaring the moduli of both sides of (56), integrating around the circle z=r, and using Parseval’s theorem, we note that(57)∑m=1n2m−1−−1m2Am2r2m+∑m=n+1∞cm2r2m<∑m=1n−1A1−−1m−2Bm2Am2r2m.
Therefore,(58)∑m=1n2m−1−−1m2Am2r2m≤∑m=1n−1A1−−1m−2Bm2Am2r2m.
Taking limit asr⟶1−, we obtain the required result.
In the following theorem,erfx for an arbitrary x denotes the error function and we need an Euler integral representation for the special class of hypergeometric functions given in [28] and defined as follows.
ForRec>Reb>0,(59)3F2a,b2,b+12;c2,c+12;x=ΓcΓbΓc−b∫01tb−11−tc−b−11−xt2−adt.Theorem 7.
Letf∈Sμ,b,csA,B. Then, for z=r0<r<1,(60)∧A,B,r≤ℬμ,bcfz≤∨A,B,r,where(61)∨A,B,r=rBAF23B−A2B,12,1;1,32;−Br2+B−A∑k=0∞−Br2kk+1F23B−A2B,k+12,k+22;k+22,k+32;−Br2,B≠0,−π2Aerf−A2r+expAr22−1,B=0,A<0,∧A,B,r=rB3F2B−A2B,12,1;1,32;Br2+B−A∑k=0∞Br2kk+1F23B−A2B,k+12,k+22;k+22,k+32;Br2B≠0,π2AerfA2r+exp−Ar22−1,B=0,A>0.
Equality is obtained for the function(62)ℬμ,bcfz=∫0z1+Byv2−A−B/2B1+Ayv1+Byvdv,if z=r0<r<1 and y=±1.Proof.
Sincef∈Sμ,b,csA,B, then(63)ℬμ,bcfz′=Φμ,bcfzpz,where Φμ,bcfz∈S∗A,B by Theorem 1 and p≺1+Az/1+Bz. Using (41) and subordination property, we have that(64)Φμ,bcfz≤r1+Br2B−A/2B,B≠0,rexpAr22,B=0,(65)Φμ,bcfz≥r1−Br2B−A/2B,B≠0,rexp−Ar22,B=0.
By adopting (13) and (64), we obtain(66)ℬμ,bcfz≤∫0rΦμ,bcfρeiθpρeiθρdρ≤∫0r1+Bρ2−B−A/2B1+Aρ1+Bρdρ=AB∫0r1+Bρ2−B−A/2Bdρ+B−AB∑k=0∞−Bk∫0rρk1+Bρ2−B−A/2B=ArB∫011+Br2u2−B−A/2Bdu+B−ArB∑k=0∞−Brk∫0ruk1+Br2u2−B−A/2Bdu.
Applying hypergeometric function (59), we obtain the upper bound for the case B≠0. In case B=0, applying (13) and using (64) in (66), we obtain(67)ℬμ,bcfz≤∫0rexpAρ22dρ+∫0rAρexpAρ22dρ=−2A∫0−A/2rexp−u2du+expAr22−1,Aρ22=−u2,A<0=−π2Aerf−A2r+expAr22−1.
This establishes the upper bound. To prove the lower bound, we consider a pointz0z0=r<1 such that ℬμ,bcfz≥ℬμ,bcfz0∀z:z=r. Let C be an arc in U which is mapped by the function w=ℬμ,bcfz onto a line segment L connecting origin to the point ℬμ,bcfz0 and lying completely in the image of U under ℬμ,bcfz. Thus, by (13) and (65), we obtain(68)ℬμ,bcfz≥ℬμ,bcfz0=∫Ldw=∫Cℬμ,bcfz′dz≥∫0r1−Bρ2−B−A/2B1−Aρ1−Bρdρ.
Adopting similar procedures as used in finding the upper bounds from (65), we obtain the desired result.
ForA=1andB=−1, we obtain a more reduced form of Theorem 7 which is contained in the following corollary.Corollary 9.
Iff∈Sμ,b,cs1,−1, then(69)ln1+r1+r2≤ℬμ,bcfz≤r1−rr<1.
This bound cannot be improved.Corollary 10.
Iff∈Sμ,b,cs0,−1, then(70)ln1+r1+r21/2≤ℬμ,bcfz≤ln1+r1−r1/4+r21−rr<1.
This bound is sharp.Remark 2.
In respect of the lower bounds ofℬμ,bcfz for the classes Sμ,b,cs1,−1 and Sμ,b,cs0,−1, given by (69) and (70), respectively, we note that the disc of the maximum radius is contained in the image domain ℬμ,bcfUr if f∈Sμ,b,cs1,−1 and f∈Sμ,b,cs0,−1, respectively, where Urz∈ℂ:z=r<1.
In view of Remark2, we note that as r⟶1− in the lower bound of both Corollaries 9 and 10, we have the following results giving the omission values for the classes Sμ,b,cs1,−1 and Sμ,b,cs0,−1.Corollary 11.
Letf∈Sμ,b,cs1,−1 and w∈ℂ be such that fz≠wz∈U. Then, w>0.34657359.Corollary 12.
Letf∈Sμ,b,cs0,−1 and w∈ℂ be such that fz≠wz∈U. Then, w>0.173286795.


## 4. Conclusion

Bessel functions are essential in many branches of mathematics and applied mathematics. Recently, there has been a clear interest on Bessel and hypergeometric functions from the point of view of geometric function theory. As a result, we presented some subclasses of analytic functions with respect to symmetrical points, which were associated with Bessel function. The geometric properties of these aforementioned classes which include integral-preserving properties, coefficients, and distortion results were studied. As a consequence of our investigation, some relevant special cases were pointed out. In addition, to capture more new results under the current examination, new idea and applications can be investigated with some positive and novel outcomes in various fields of science, especially in mathematics. These new investigations will be presented in future research work being processed by the authors of the present article.



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