# Uniformly Alpha-Quasi-Convex Functions Defined by Janowski Functions

Authors: Shahid Mahmood; Sarfraz Nawaz Malik; Sumbal Farman; S. M. Jawwad Riaz; Shabieh Farwa
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this work, we aim to introduce and study a new subclass of analytic functions related to the oval and petal type domain. This includes various interesting properties such as integral representation, sufficiency criteria, inclusion results, and the convolution properties for newly introduced class.


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## Body

## 1. Introduction and Preliminaries

LetA be the class of functions fz of the form(1)fz=z+∑n=2∞anzn,which are analytic in the open unit disk U=z:z<1. Let S, S∗γ, Cγ, and Kγ be the subclasses of A consisting of functions which are univalent, starlike of order γ, convex of order γ, and close-to-convex of order γ, respectively, 0≤γ<1. Clearly S∗0=S∗, the class of starlike univalent functions, C0=C, the class of convex univalent functions, and K0=K, the class of close-to-convex univalent functions.For analytic functionsfz=z+∑n=2∞anzn and gz=z+∑n=2∞bnzn, the convolution or Hadamard product is denoted by f∗g and is defined by (2)f∗gz=z+∑n=2∞anbnzn.A functionfz is said to be subordinate to the function gz, written symbolically as fz≺gz, if there exists a Schwarz function wz such that (3)fz=gwz,z∈U,where w0=0, wz<1 for z∈U. The class PA,B consists of functions fz which are analytic in U with f0=1 and (4)fz≺1+Az1+Bz,where -1≤B<A≤1. If a function fz belongs to PA,B, then it maps U on to the domain ΩA,B, defined by (5)ΩA,B=w:w-1-AB1-B2<A-B1-B2. The domain ΩA,B represents an open circular disk, having center at real axis and whose diameter points are d1=(1-A)/(1-B) and d2=(1+A)/(1+B). For B=-1 and A=1, the class PA,B reduces to the well-known class P of Carathèodory functions; see [1]. The class PA,B is connected with the class P of Carathèodory functions by the following relation. (6)fz∈P⟺A+1fz-A-1B+1fz-B-1∈PA,B. Janowski [2] introduced the classes S∗A,B and CA,B and studied them comprehensively. He established the Alexander type relation between these classes.In 1999, Kanas and Wiśniowska [3] gave the concept of general conic domain by introducing a domain Ωβ, defined by (7)Ωβ=u+iv:u>βu-12+v2. The domain Ωβ represents a right half plane when β=0, the hyperbolic regions for 0<β<1, a parabolic region for β=1, and it represents elliptic regions when β>1. The extremal function for conic domain Ωβ is the mapping pβz with pβ0=1 and pβ′0>0 and is defined by(8)pβz=1+z1-z,β=0,1+2π2log⁡1+z1-z2,β=1,1+21-β2sin⁡h22πarccos⁡βarctan⁡hz,0<β<1,1+1β2-1sin⁡π2Xt∫0uz/t11-x21-tx2dx+1β2-1,β>1,where uz=(z-t)/(1-tz), t∈0,1, z∈U, and z is chosen such that β=coshπX′(t)/4X(t), Xt is Legendre’s complete elliptic integral of first kind, and X′t is complementary integral of Xt; for more details, see [3]. Meanwhile, Kanas and Wiśniowska [3, 4] introduced a class Ppβ which maps U on to the conic regions, defined by the domain Ωβ. They also defined the classes β-UCV and β-ST of β-uniformly convex functions and their corresponding β-starlike functions; see [3, 4]. Recently Noor and Malik [5] introduced new geometrical structures of oval and petal type shape as image domain and defined the class β-PA,B of functions which give these types of mappings. The concepts of Janowski functions and conic domains are combined together to define the class β-PA,B which represents the oval and petal type regions as image domain.Definition 1 (see [5]).
A functionpz∈β-PA,B if and only if, (9)pz≺pβA,B;z,β≥0,-1≤B<A≤1,where(10)pβA,B;z=A+1pβz-A-1B+1pβz-B-1with pβz as defined in (8).Graphically, the functionpz∈β-PA,B takes all the values in the domain ΩβA,B, -1≤B<A≤1, which is defined as (11)ΩβA,B=wz:Re⁡B-1wz-A-1B+1wz-A+1>βB-1wz-A-1B+1wz-A+1-1or, equivalently, (12)ΩβA,B=u+iv:B2-1u2+v2-2AB-1u+A2+12>-2B+1u2+v2+2A+B+2u-2A+12.Noor and Malik [5] also introduced two new classes β-UCVA,B and β-STA,B related to the domain ΩβA,B whose geometry is oval and petal type regions subject to the values of A, B, and beta. These classes are defined as follows.Definition 2 (see [5]).
A functionfz∈A is said to be in the class β-UCVA,B,  k≥0,  -1≤B<A≤1, if and only if(13)Re⁡B-1zf′z′/f′z-A-1B+1zf′z′/f′z-A+1>βB-1zf′z′/f′z-A-1B+1zf′z′/f′z-A+1-1,or, equivalently,(14)zf′z′f′z∈β-PA,B.Definition 3 (see [5]).
A functionfz∈A is said to be in the class β-STA,B,  k≥0,  -1≤B<A≤1, if and only if (15)Re⁡B-1zf′z/fz-A-1B+1zf′z/fz-A+1>βB-1zf′z/fz-A-1B+1zf′z/fz-A+1-1,or, equivalently,(16)zf′zfz∈β-PA,B.Motivated from the recent work presented by Noor and Malik [5], we define the following new subclass of analytic functions associated with domain ΩβA,B.Definition 4.
A functionfz∈A is said to be in the class β-UQαA,B,C,D, β≥0, 0≤α≤1, -1≤B<A≤1, -1≤D<C≤1, if and only if(17)Re⁡B-1wz-A-1B+1wz-A+1>βB-1wz-A-1B+1wz-A+1-1,or (18)wz∈β-PA,B,where(19)wz=1-αf′zg′z+αzf′z′g′zfor some gz∈β-UCVC,D.It follows from the above definition thatfz∈β-UQαA,B,C,D if and only if(20)Fz=1-αfz+αzf′z∈β-UKA,B,C,D,where β-UKA,B,C,D is the class of uniformly Janowski close-to-convex functions, introduced and studied by Mahmood et al. [6].Special Cases. By setting certain values to different parameters, the class β-UQαA,B,C,D coincides with several renowned classes of analytic functions, as given below.(1)
β-UQ0A,B,C,D=β-UKA,B,C,D, β-UQ1A,B,C,D=β-UQA,B,C,D are well-known classes of uniformly Janowski close-to-convex and uniformly Janowski quasi-convex functions respectively, introduced and studied by Mahmood et al. [6].(2)
β-UQα1,-1,1,-1=β-UK is the well known class of uniformly close-to-convex functions, introduced and studied by Noor et al. [7].(3)
0-UQα1,-1,1,-1=Qα is the well known class of alpha-quasi-convex functions, introduced and studied by Noor and Aboudi [8].(4)
0-UQαA,B,1,-1=QαA,B is the well known class of alpha Janowski convex functions, introduced and studied by Selvaraj and Thirupathi [9].Throughout this paper, we assume thatβ≥0, -1≤B<A≤1, -1≤D<C≤1, and 0≤α≤1 unless otherwise stated.


## 2. A Set of Lemmas

To prove our main results we need the following Lemmas.Lemma 5 (see [5]).
Letpz=1+∑n=1∞cnzn∈β-PA,B. Then (21)cn≤δA,B,β,where(22)δA,B,β=A-Bδβ2,(23)δβ=8cos-1⁡β2π21-β2,0≤β<1,8π2,β=1,π24tβ2-1R2t1+t,β>1.Lemma 6 (see [5]).
Letgz∈β-CVC,D with β≥0 and be given by (24)gz=z+∑n=2∞bnzn.Then (25)bn≤1n∏j=0n-2C-Dδβ-2jD2j+1,where δβ is defined by (23).Lemma 7 (see [10]).
Letfz and gz be in the classes C and S∗, respectively. Then for every function Fz analytic in U with F0=1, we have (26)fz∗gzFzfz∗gz∈co⁡¯FU,z∈U, where co⁡¯FU denotes the closed convex hull FU.Lemma 8 (see [6]).
Letfz∈β-UKA,B,C,D and φz∈C. Then φ∗fz∈β-UKA,B,C,D.


## 3. Main Results

Theorem 9.
Letfz∈β-UQαA,B,C,D. Then, the function Fz is defined such that(27)fz=1αz1-1/α∫0zt2-1/αFtdt,α>0,z∈Ubelongs to the class β-UKA,B,C,D.Proof.
From (27), we can write (28)1-αfz+αzf′z=Fz,z∈Uand the result follows by using (20).Theorem 10.
A functionfz having the form (1) is in the class β-UQαA,B,C,D if it satisfies(29)2β+1∑n=2∞∏j=0n-2δβC-D-2jD2j+1+∑n=2∞n1+αn-1an+∑n=2∞B+1n1+αn-1an+∑n=2∞∏j=0n-2δβC-D-2jD2j+1A+1<B-A.Proof.
Assuming that (29) holds true, then it is sufficient to show that(30)βB-1wz-A-1B+1wz-A+1-1-ReB-1wz-A-1B+1wz-A+1-1<1.Now consider(31)βB-11-αf′z/g′z+αzf′z′/g′z-A-1B+11-αf′z/g′z+αzf′z′/g′z-A+1-1-Re⁡B-11-αf′z/g′+αzf′z′/g′-A-1B+11-αf′z/g′z+αzf′z′/g′z-A+1-1≤β+1B-1f′z/g′z+αzf′′z/g′z-A-1B+1f′z/g′z+αzf′′z/g′z-A+1-1=β+1B-1f′z+αzf′′z-A-1g′zB+1f′z+αzf′′z-A+1g′z-1=β+1B-1f′z+αzf′′z-A-1g′z-B+1f′z+αzf′′z+A+1g′zB+1f′z+αzf′′z-A+1g′z=2β+1g′z-f′z-αzf′′zB+1f′z+αzf′′z-A+1g′z≤2β+11+∑n=2∞nbnzn-1-1-∑n=2∞nanzn-1-α∑n=2∞nn-1anzn-1B+11+∑n=2∞nanzn-1-α∑n=2∞nn-1anzn-1-A+11+∑n=2∞nbnzn-1=2β+1∑n=2∞nbnzn-1-∑n=2∞n1+αn-1anzn-1B-A+B+1∑n=2∞n1+αn-αanzn-1-A+1∑n=2∞nbnzn-1≤2β+1∑n=2∞nbn+∑n=2∞n1+αn-1anB-A-B+1∑n=2∞n1+αn-αanzn-1-A+1∑n=2∞nbnzn-1≤2β+1∑n=2∞nbn+∑n=2∞n1+αn-1anB-A-B+1∑n=2∞n1+αn-1an-A+1∑n=2∞nbn≤2β+1∑n=2∞∏j=0n-2δβC-D-2jD/2j+1+∑n=2∞n1+αn-1anB-A-B+1∑n=2∞n1+αn-αan-A+1∑n=2∞∏j=0n-2δβC-D-2jD/2j+1.In the last inequality, we have used the following coefficient inequality for functions belonging to the class β-CVC,D (see [5]):(32)bn≤1n∏j=0n-2δβC-D-2jD2j+1.
The last inequality is bounded above by 1, if(33)2β+1∑n=2∞∏j=0n-2δβC-D-2jD2j+1+∑n=2∞n1+αn-1an+∑n=2∞B+1n1+αn-1an+∑n=2∞∏j=0n-2δβC-D-2jD2j+1A+1<B-A.When α=0, Theorem 10 reduces to the following known result, proved in [6].Corollary 11.
If a functionfz∈A satisfies(34)2β+1∑n=2∞∏j=0n-2δβC-D-2jD2j+1+∑n=2∞nan+∑n=2∞B+1nan+∑n=2∞∏j=0n-2δβC-D-2jD2j+1A+1<B-A,then fz∈β-UKA,B,C,D.Whenα=1, Theorem 10 takes the form of following known result, proved in [6].Corollary 12.
A functionfz∈A is said to be in the class β-UQA,B,C,D if it satisfies the condition (35)2β+1∑n=2∞∏j=0n-2δβC-D-2jD2j+1+∑n=2∞n2an+∑n=2∞B+1n2an+∑n=2∞∏j=0n-2δβC-D-2jD2j+1A+1<B-A.Theorem 13.
Letfz∈β-UQαA,B,C,D and be of the form (1). Then for n≥2, we have (36)an≤1n+αnn-1∏j=0n-2δβC-D-2jD2j+1+δA,B,β∑j=1n-1∏m=0j-2δβC-D-2mD2m+1,where δA,B,β is given by (22).Proof.
By definition, forfz∈β-UQαA,B,C,D, we get(37)1-αf′zg′z+αzf′z′g′z=pz,where (38)pz=1+∑n=1∞cnzn≺pβA,B;z.We can write (37) as(39)f′z-αzf′′z=pzg′z.Now using the series expansion, (39) reduces to (40)1+∑n=2∞nanzn-1+α∑n=2∞nn-1anzn-1=1+∑n=2∞nbnzn-11+∑n=2∞cn-1zn-1.
Equating coefficients ofzn-1 on both sides, one may get (41)nan+αnn-1an=nbn+∑j=1n-1jbjcn-j.
This implies that(42)ann+αnn-1≤nbn+∑j=1n-1jbjcn-j.
Now applying Lemmas5 and 6, (42) reduces to(43)an≤1n+αnn-1∏j=0n-2δβC-D-2jD2j+1+δA,B,β∑j=1n-1∏m=0j-2δβC-D-2mD2m+1,where δA,B,β is given by (22). This completes the proof.By settingα=0 in Theorem 13, we have the following known result, proved by Mahmood et al. [6].Corollary 14.
Letfz∈β-UKA,B,C,D and an be the nth coefficient of Taylor series of fz. Then, for n≥2, we obtain (44)an≤1n∏j=0n-2δβC-D-2jD2j+1+δA,B,β∑j=1n-1∏m=0j-2δβC-D-2mDA-Bδβ2m+1,where δA,B,β is given by (22).Forα=1, Theorem 13 takes the form of following known result, proved by Mahmood et al. [6].Corollary 15.
Let the functionfz of the form (1) belong to the class β-UQA,B,C,D. Then, (45)an≤1n2∏j=0n-2δβC-D-2jD2j+1+δA,B,β∑j=1n-1∏m=0j-2δβC-D-2mDA-Bδβ2m+1, where n≥2 and δA,B,β is given by (22).When we considerA=C=1, B=D=-1, and α=0 in Theorem 13, we obtain the following known result, proved by Noor et al. [7].Corollary 16.
Letfz∈β-UK and an be the nth coefficient of Taylor series of fz. Then, (46)an≤δβn-1n!+δβn∑j=0n-1δβj-1j-1!,n≥2.By Puttingβ=0, C=1, and D=-1 in Theorem 13, we have the following known result proved by Selvaraj and Thirupathi [9].Corollary 17.
Letfz∈QαA,B and be of the form (1). Then, (47)an≤11+αn-11+A-Bn-12,n≥2.When we takeA=C=1, B=D=-1, and β=0 in Theorem 13, we obtain the following known result proved by Noor and Aboudi [8].Corollary 18.
Letfz∈QαA,B and an be the nth coefficient of Taylor series of fz. Then, (48)an≤n1+αn-1,n≥2.Theorem 19.
Letfz∈β-UQαA,B,C,D, α≥0, and ϕz be any convex univalent function in U. Then (49)fz∗ϕz∈β-UQαA,B,C,D,z∈U.Proof.
Letfz∈β-UQαA,B,C,D, α≥0. Then by (20), we have (50)1-αfz+αzf′z∈β-UKA,B,C,D.Using Lemma 8, one may have (51)1-αfz+αzf′z∗ϕz∈β-UKA,B,C,D,which implies that (52)1-αf∗ϕz+αzf∗ϕ′z∈β-UKA,B,C,D.Hence by (20), fz∗ϕz∈β-UQαA,B,C,D. This completes the proof.Applications of Theorem19Theorem 20.
Letfz∈β-UQαA,B,C,D. Then the functions ϕiz, such that (53)ϕ1z=∫0zfttdt,ϕ2z=2z∫0zftdt,ϕ3z=∫0zft-ftt-txdt,x≤1,x≠1,ϕ4z=λ+1zλ∫0ztλ-1ftdt,Re⁡λ>0z∈U,belong to the class β-UQαA,B,C,D.Proof.
The proof follows immediately, when we observe that(54)ϕiz=f∗ψiz,i=1,2,3,4z∈U,where ψiz is convex in U such that (55)ψ1=-log⁡1-z=∑n=1∞1nzn,ψ2=-z+log⁡1-zz,ψ3=11-xlog⁡1-xz1-z,ψ4=∑n=1∞1+λn+λzn,Re⁡λ>0,z∈U.Theorem 21.
Forα≥0, we have β-UQαA,B,C,D⊆β-UKA,B,C,D.Proof.
Forα=0, the result is obvious. So we assume that α>0 and using the integral representation for fz∈β-UQαA,B,C,D, we have(56)fz=1αz1-1/α∫0zt2-1/αFtdt,z∈U,where Fz∈β-UKA,B,C,D, since fz=ϕαz∗Fz, where (57)ϕαz=∑n=1∞1αn-1+1zn,z∈Uis a convex function in U. Now using Lemma 8, we conclude that fz∈β-UKA,B,C,D. This completes the proof.Theorem 22.
If0≤α1≤α2, Then, (58)β-UQα2A,B,C,D⊂β-UQα1A,B,C,D.Proof.
Letfz∈β-UQα2A,B,C.D. Then (59)1-α2f′zg′z+α2zf′z′g′z∈β-PA,B.Now consider (60)jα1,f;z=1-α1f′zg′z+α1zf′z′g′z=1-α1α2f′zg′z+α1α21-α2f′zg′z+α2zf′z′g′z=1-α1α2j0,f;z+α1α2jα2,f;z.Since β-PA,B is convex, (61)1-α1α2j0,f;z+α1α2jα2,f;z∈β-PA,B and consequently fz∈β-UQα1A,B,C,D. This completes the proof.



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