# On Subclasses of Uniformly Spiral-like Functions Associated with Generalized Bessel Functions

Authors: B. A. Frasin; Ibtisam Aldawish
Journal: Journal of Function Spaces
 (2019)

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## Abstract

The main object of this paper is to find necessary and sufficient conditions for generalized Bessel functions of first kindzup(z) to be in the classes SPp(α,β) and UCSP(α,β) of uniformly spiral-like functions and also give necessary and sufficient conditions for z(2-up(z)) to be in the above classes. Furthermore, we give necessary and sufficient conditions for I(κ,c)f to be in UCSPT(α,β) provided that the function f is in the class Rτ(A,B). Finally, we give conditions for the integral operator G(κ,c,z)=∫0z(2-up(t))dt to be in the class UCSPT(α,β). Several corollaries and consequences of the main results are also considered.


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## Body

## 1. Introduction and Definitions

LetA denote the class of the normalized functions of the form(1)fz=z+∑n=2∞anzn,which are analytic in the open unit disk U={z∈C:z<1}. Further, let T be a subclass of A consisting of functions of the form,(2)fz=z-∑n=2∞anzn,z∈U.A function f∈A is spiral-like if (3)Re-iαzf′zfz>0,for some α with α<π/2and for all z∈U. Also f(z) is convex spiral-like if zf′(z) is spiral-like.In [1], Selvaraj and Geetha introduced the following subclasses of uniformly spiral-like and convex spiral-like functions.Definition 1.
A functionf of the form (1) is said to be in the class SPp(α,β) if it satisfies the following condition: (4)Re-iαzf′zfz>zf′zf′z-1+βz∈U;α<π2;0≤β<1and f∈UCSP(α,β) if and only if zf′(z)∈SPp(α,β).We write(5)SPpTα,β=SPpα,β∩T,UCSPTα,β=UCSPα,β∩T.In particular, we note thatSPp(α,0)=SPp(α) and UCSP(α,0)=UCSP(α), the classes of uniformly spiral-like and uniformly convex spiral-like were introduced by Ravichandran et al. [2]. For α=0, the classes UCSP(α) and SPp(α), respectively, reduce to the classes UCV and SP introduced and studied by Ronning [3].For more interesting developments of some related subclasses of uniformly spiral-like and uniformly convex spiral-like, the readers may be referred to the works of Frasin [4, 5], Goodman [6, 7], Tariq Al-Hawary and Frasin [8], Kanas and Wisniowska [9, 10] and Ronning [3, 11].A functionf∈A is said to be in the class Rτ(A,B),τ∈C∖{0}, -1≤B<A≤1, if it satisfies the inequality (6)f′z-1A-Bτ-Bf′z-1<1,z∈U.This class was introduced by Dixit and Pal [12].The generalized Bessel functionwp (see, [13]) is defined as a particular solution of the linear differential equation (7)zw′′z+bzw′z+cz2-p2+1-bpwz=0,where b,p,c∈C. The analytic function wp has the form (8)wpz=∑n=0∞-1ncnn!Γp+n+b+1/2.z22n+p,z∈C.Now, the generalized and normalized Bessel function up is defined with the transformation (9)upz=2pΓp+n+b+12z-p/2wpz1/2=∑n=0∞-c/4nκnn!zn,where κ=p+(b+1)/2≠0,-1,-2,… and (a)n is the well-known Pochhammer (or Appell) symbol, defined in terms of the Euler Gamma function for a≠0,-1,-2,… by(10)an=Γa+nΓa=1,ifn=0aa+1a+2…a+n-1,ifn∈N.The function up is analytic on C and satisfies the second-order linear differential equation (11)4z2u′′z+22p+b+1zu′z+czuz=0.Using the Hadamard product, we now considered a linear operator I(κ,c):A→A defined by (12)Iκ,cf=zupz∗fz=z+∑n=2∞-c/4n-1κn-1n-1!anzn,where ∗ denote the convolution or Hadamard product of two series.The study of the generalized Bessel function is a recent interesting topic in geometric function theory. We refer, in this connection, to the works of [13–15] and others.Motivated by results on connections between various subclasses of analytic univalent functions by using hypergeometric functions (see, for example, [16–20])), and the work done in [21–24], we determine necessary and sufficient conditions for zup(z) to be in SPp(α,β) and UCSP(α,β) and also give necessary and sufficient conditions for z(2-up(z)) to be in the function classes SPpT(α,β) and UCSPT(α,β). Furthermore, we give necessary and sufficient conditions for I(κ,c)f to be in UCSPT(α,β) provided that the function f is in the class Rτ(A,B). Finally, we give conditions for the integral operator G(κ,c,z)=∫0z‍(2-up(t))dt to be in the class UCSPT(α,β).To establish our main results, we need the following Lemmas.Lemma 2 (see [1]).
(i) A sufficient condition for a functionf of the form (1) to be in the class SPp(α,β) is that(13)∑n=2∞2n-cos⁡α-βan≤cos⁡α-βα<π/2;0≤β<1and a necessary and sufficient condition for a function f of the form (2) to be in the class SPpT(α,β) is that condition (13) is satisfied. In particular, when β=0, we obtain a sufficient condition for a function f of the form (1) to be in the class SPp(α) is that(14)∑n=2∞2n-cos⁡αan≤cos⁡αα<π2and a necessary and sufficient condition for a function f of the form (2) to be in the class SPpT(α) is that condition (14) is satisfied.
(ii) A sufficient condition for a functionf of the form (1) to be in the class UCSP(α,β) is that(15)∑n=2∞n2n-cos⁡α-βan≤cos⁡α-βα<π2;0≤β<1and a necessary and sufficient condition for a function f of the form (2) to be in the class UCSPT(α,β) is that condition (15) is satisfied. In particular, when β=0, we obtain a sufficient condition for a function f of the form (1) to be in the class UCSP(α) is that(16)∑n=2∞n2n-cos⁡αan≤cos⁡αα<π2and a necessary and sufficient condition for a function f of the form (2) to be in the class UCSPT(α) is that condition (16) is satisfied.Lemma 3 (see [12]).
Iff∈Rτ(A,B) is of the form (1), then (17)an≤A-Bτn,n∈N-1.
The result is sharp for the function(18)fz=∫0z1+A-Bτtn-11+Btn-1dt,z∈U;n∈N-1.Lemma 4 (see [15]).
Ifb,p,c∈C and κ≠0,-1,-2,…, then the function up satisfies the recursive relations (19)up′z=-c/4κup+1z,up′′z=-c/42κκ+1up+2z,for all z∈C.


## 2. The Necessary and Sufficient Conditions

Unless otherwise mentioned, we shall assume in this paper thatα<π/2 and 0≤β<1.First we obtain the necessary condition forzup to be in SPp(α,β).Theorem 5.
Ifc<0, κ>0(κ≠0,-1,-2,…), then zup is in SPp(α,β) if(20)2up′1+2-cos⁡α-βup1-1≤cos⁡α-β.Proof.
Since(21)zuPz=z+∑n=2∞-c/4n-1κn-1n-1!zn,according to (13), we must show that(22)∑n=2∞2n-cos⁡α-β-c/4n-1κn-1n-1!≤cos⁡α-β.⁡Writing(23)n=n-1+1,we have(24)∑n=2∞2n-cos⁡α-β-c/4n-1κn-1n-1!=2∑n=2∞n-1-c/4n-1κn-1n-1!+∑n=2∞2-cos⁡α-β-c/4n-1κn-1n-1!=2∑n=2∞-c/4n-1κn-1n-2!+∑n=2∞2-cos⁡α-β-c/4n-1κn-1n-1!=2∑n=0∞-c/4n+1κn+1n!+2-cos⁡α-β∑n=0∞-c/4n+1κn+1n+1!=2-c/4κ∑n=0∞-c/4nκ+1nn!+2-cos⁡α-β∑n=0∞-c/4n+1κn+1n+1!=2-c/4κup+11+2-cos⁡α-βup1-1=2up′1+2-cos⁡α-βup1-1.But this last expression is bounded above by cos⁡α-β if (20) holds.Corollary 6.
Ifc<0, κ>0(κ≠0,-1,-2,…), then z(2-up(z)) is in SPpT(α,β) if and only if the condition (20) is satisfied.Proof.
Since(25)z2-upz=z-∑n=2∞-c/4n-1κn-1n-1!zn.
By using the same techniques given in the proof of Theorem5, we have Corollary 6.Theorem 7.
Ifc<0, κ>0(κ≠0,-1,-2,…), then zup is in SPp(α,β) if(26)e-c/4κ-c2κ+2-cos⁡α-β1-ec/4κ≤cos⁡α-β.Proof.
We note that(κ)n-1=κ(κ+1)(κ+2)⋯(κ+n-2)≥κ(κ+1)n-2≥κn-1,(n∈N). From (24), we get (27)∑n=2∞2n-cos⁡α-β-c/4n-1κn-1n-1!≤2∑n=2∞n-1-c4κn-1n-1!+2-cos⁡α-β∑n=2∞-c/4κn-1n-1!=-c2κe-c/4κ+2-cos⁡α-βe-c/4κ-1.
Therefore, we see that the last expression is bounded above bycos⁡α-β if (26) is satisfied.Corollary 8.
Ifc<0, κ>0(κ≠0,-1,-2,…), then z(2-up(z)) is in SPpT(α,β) if and only if the condition (26) is satisfied.Theorem 9.
Ifc<0, κ>0(κ≠0,-1,-2,…), then zup(z) is in UCSP(α,β) if(28)2up′′1+6-cos⁡α-βup′1+2-cos⁡α-βup1-1≤cos⁡α-β.Proof.
In view of (15), we must show that(29)∑n=2∞n2n-cos⁡α-β-c/4n-1κn-1n-1!≤cos⁡α-β.
Writing(30)n=n-1+1,n2=n-1n-2+3n-1+1.Thus, we have (31)∑n=2∞n2n-cos⁡α-β-c/4n-1κn-1n-1!=2∑n=2∞n-1n-2-c/4n-1κn-1n-1!+6-cos⁡α-β∑n=2∞n-1-c/4n-1κn-1n-1!+2-cos⁡α-β∑n=2∞-c/4n-1κn-1n-1!.=2∑n=3∞-c/4n-1κn-1n-3!+6-cos⁡α-β∑n=2∞-c/4n-1κn-1n-2!+2-cos⁡α-β∑n=2∞-c/4n-1κn-1n-1!.=2-c/42κκ+1∑n=0∞-c/4nκ+2nn!+6-cos⁡α-β-c/4κ∑n=0∞-c/4nκ+1nn!+2-cos⁡α-β∑n=0∞-c/4n+1κn+1n+1!=2up′′1+6-cos⁡α-βup′1+2-cos⁡α-βup1-1.But this last expression is bounded above by cos⁡α-β if (28) holds.By using a similar method as in the proof of Corollary6, we have the following result.Corollary 10.
Ifc<0, κ>0(κ≠0,-1,-2,…), then z(2-up(z)) is in UCSPT(α,β) if and only if the condition (28) is satisfied.The proof of Theorem11 (below) is much akin to that of Theorem 7, and so the details may be omitted.Theorem 11.
Ifc<0, κ>0(κ≠0,-1,-2,…), then z(2-up(z)) is in UCSPT(α,β) if and only if(32)e-c/4κc28κ+6-cos⁡α-β-c4κ+2-cos⁡α-β1-ec/4κ≤cos⁡α-β.


## 3. Inclusion Properties

Making use of Lemma3, we will study the action of the Bessel function on the class UCSPT(α,β).Theorem 12.
Letc<0, κ>0(κ≠0,-1,-2,…). If f∈Rτ(A,B),then I(κ,c)f is in UCSPT(α,β) if and only if(33)A-Bτ2up′1+2-cos⁡α-βup1-1≤cos⁡α-β.Proof.
In view of (15), it suffices to show that (34)∑n=2∞n2n-cos⁡α-β-c/4n-1κn-1n-1!an≤cos⁡α-β.
Sincef∈Rτ(A,B), then by Lemma 3, we get(35)an≤A-Bτn.
Thus, we must show that(36)∑n=2∞n2n-cos⁡α-β-c/4n-1κn-1n-1!an≤A-Bτ∑n=2∞2n-cos⁡α-β-c/4n-1κn-1n-1!.
The remaining part of the proof of Theorem12 is similar to that of Theorem 5, and so we omit the details.


## 4. An Integral Operator

In this section, we obtain the necessary and sufficient conditions for the integral operatorG(κ,c,z) defined by(37)Gκ,c,z=∫0z2-uptdtto be in UCSPT(α,β).Theorem 13.
If<0, κ>0(κ≠0,-1,-2,…), then the integral operator G(κ,c,z) is in UCSPT(α,β) if and only if the condition (20) is satisfied.Proof.
Since(38)Gκ,c,z=z-∑n=2∞-c/4n-1κn-1znn!then, in view of (15), we need only to show that (39)∑n=2∞n2n-cos⁡α-β-c/4n-1κn-1n!≤cos⁡α-β
or equivalently(40)∑n=2∞2n-cos⁡α-β-c/4n-1κn-1n-1!≤cos⁡α-β.
The remaining part of the proof is similar to that of Theorem5, and so we omit the details.The proofs of Theorems14 and 15 are much akin to that of Theorem 7, and so the details may be omitted.Theorem 14.
Letc<0, κ>0(κ≠0,-1,-2,…). If f∈Rτ(A,B),then I(κ,c)f is in UCSPT(α,β) if and only if(41)A-Bτe-c/4κ-c2κ+2-cos⁡α-β1-ec/4κ≤cos⁡α-β.Theorem 15.
If<0, κ>0(κ≠0,-1,-2,…), then the integral operator G(κ,c,z) is in UCSPT(α,β) if and only if the condition (32) is satisfied.


## 5. Corollaries and Consequences

In this section, we apply our main results in order to deduce each of the following corollaries and consequences.Corollary 16.
Ifc<0, κ>0(κ≠0,-1,-2,…),then zup is in SPp(α) if(42)2up′1+2-cos⁡αup1-1≤cos⁡α.Corollary 17.
Ifc<0, κ>0(κ≠0,-1,-2,…), then z(2-up(z)) is in SPpT(α) if and only if the condition (42) is satisfied.Corollary 18.
Ifc<0, κ>0(κ≠0,-1,-2,…), then zup is in SPp(α) if(43)e-c/4κ-c2κ+2-cos⁡α1-ec/4κ≤cos⁡α.Corollary 19.
Ifc<0, κ>0(κ≠0,-1,-2,…), then z(2-up(z)) is in SPpT(α) if and only if the condition (43) is satisfied.Corollary 20.
Ifc<0, κ>0(κ≠0,-1,-2,…), then zup(z) is in UCSP(α) if(44)2up′′1+6-cos⁡αup′1+2-cos⁡αup1-1≤cos⁡α.Corollary 21.
Ifc<0, κ>0(κ≠0,-1,-2,…), then z(2-up(z)) is in UCSPT(α) if and only if(45)e-c/4κc28κ+6-cos⁡α-c4κ+2-cos⁡α1-ec/4κ≤cos⁡α.Corollary 22.
Letc<0, κ>0(κ≠0,-1,-2,…). If f∈Rτ(A,B),then I(κ,c)f is in UCSPT(α) if and only if (46)A-Bτ2up′1+2-cos⁡αup1-1≤cos⁡α.Corollary 23.
If<0, κ>0(κ≠0,-1,-2,…), then the integral operator G(κ,c,z) is in UCSPT(α) if and only if the condition (42) is satisfied.Corollary 24.
Letc<0, κ>0(κ≠0,-1,-2,…). If f∈Rτ(A,B),then I(κ,c)f is in UCSPT(α) if and only if(47)A-Bτe-c/4κ-c2κ+2-cos⁡α1-ec/4κ≤cos⁡α.Corollary 25.
If<0, κ>0(κ≠0,-1,-2,…), then the integral operator G(κ,c,z) is in UCSPT(α) if and only if the condition (45) is satisfied.Remark 26.
If we putα=0 in Corollary 6, we obtain Theorem 5 in [22] for λ=1 and β=1.



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