# Class of Analytic Function Related with Uniformly Convex and Janowski’s Functions

Authors: Akhter Rasheed; Saqib Hussain; Muhammad Asad Zaighum; Maslina Darus
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this paper, we introduce a new subclass of analytic functions in open unit disc. We obtain coefficient estimates, extreme points, and distortion theorem. We also derived the radii of close-to-convexity and starlikeness for this class.


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## Body

## 1. Introduction

LetA denote the class of normalized analytic functions in open unit disc Δ=z:z<1 and having Taylor series of form(1)fz=z+∑n=2∞anzn.Silverman [1] introduced and studied a subclass A- of A consisting of functions of the form(2)fz=z-∑n=2∞anzn,an>0,z∈Δ.A complex valued functionfz is said to be univalent in Δ if z1≠z2 implies fz1≠fz2, for all z1, z2∈Δ. Let S be the subclass of A composed of univalent functions in Δ. By ST, CV, and K, we mean the well-known subclasses of S that are starlike, convex, and close-to-convex functions, respectively; for detail see [2, 3].In 1991, Goodman [2, 3] introduced classes UCV and UST of uniformly convex and uniformly starlike functions, respectively. A function f∈S is uniformly convex if fz maps every circular arc γ contained in Δ with center ζ∈Δ onto a convex arc. The function f∈S is uniformly starlike if fz maps every circular arc ζ contained in Δ with center ζ∈Δ onto a starlike arc with respect to f(ζ). A more useful representation of UCV and UST was given in [2, 3]; see also [4–7] as(3)f∈UCV⇔f∈AandRe1+zf′′zf′z>zf′′zf′z,z∈Δ,and(4)f∈UST⇔f∈AandRezf′zfz>zf′zfz-1,z∈Δ,In 2011, Noor et. al [8] introduced and studied a class k-USTA,B, k≥0, -1≤B<A≤1 as follows.A functionfz∈A is said to be in the class k-USTA,B if and only if(5)ReB-1zf′z/fz-A-1B+1zf′z/fz-A+1>kB-1zf′z/fz-A-1B+1zf′z/fz-A+1-1.The abovementioned few classes were widely investigated by many authors in the last decades; see [4, 8–15] and the references cited therein. By taking motivation from cited work, we define a unified class of analytic functions as follows.Definition 1.
A functionfz∈A is said to be in the class k-USTA,B,δ,λ if and only if(6)ReB-1zf′z+δ1+2λz2f″z+λz3f′′′z/1-δfz+δzf′z+λz2f″z-A-1B+1zf′z+δ1+2λz2f″z+λz3f′′′z/1-δfz+δzf′z+λz2f″z-A+1>kB-1zf′z+δ1+2λz2f″z+λz3f′′′z/1-δfz+δzf′z+λz2f″z-A-1B+1zf′z+δ1+2λz2f″z+λz3f′′′z/1-δfz+δzf′z+λz2f″z-A+1-1,where0≤δ≤1, 0≤λ<1, k≥0, -1≤B<A≤1.We further letk-VUSTA,B,δ,λ=k-USTA,B,δ,λ∩A-.Special Cases (i)
k - U S T A , B , 0 , λ = k - U S T A , B [8]. (ii)
k - U S T A , B , 1,0 = k - U C V A , B [8]. (iii)
k - U S T 1 - 2 α , - 1,0 , λ = k - U S T α [16]. (iv)
k - U S T 1 - 2 α , - 1,1 , 0 = k - U C V α[16]. (v)
k - U S T 1 , - 1,0 , λ = k - U S T [14].Throughout this paper0≤δ≤1, 0≤λ<1, k≥0-, and -1≤B<A≤1, unless otherwise stated.In the next section we shall present our major findings as theorems and corollaries.


## 2. Main Results

Theorem 2.
Letf∈A be given by (1); then f is in k-USTA,B,δ,λ if(7)∑n=2∞2k+1n-1Φn+B+1Ψn-A+1Φnan≤B-A,where(8)Φn=1-δ+δn1+λn-1andΨn=n+δnn-11+λn.Proof.
Assume that inequality (7) holds true. Then we have (9)kB-11-δzf′+δzf′+1+2λz2f′′+λz3f′′′/1-δf+δzf′+λz2f′′-A-1B+11-δzf′+δzf′+1+2λz2f′′+λz3f′′′/1-δf+δzf′+λz2f′′-A+1-1-ReB-11-δzf′+δzf′+1+2λz2f′′+λz3f′′′/1-δf+δzf′+λz2f′′-A-1B+11-δzf′+δzf′+1+2λz2f′′+λz3f′′′/1-δf+δzf′+λz2f′′-A+1-1≤k+1B-11-δzf′+δzf′+1+2λz2f′′+λz3f′′′/1-δf+δzf′+λz2f′′-A-1B+11-δzf′+δzf′+1+2λz2f′′+λz3f′′′/1-δf+δzf′+λz2f′′-A+1-1=2k+11-δf-zf′-δ1+λz2f′′+λz3f′′′B+11-δzf′+δzf′+1+2λz2f′′+λz3f′′′-A+11-δf+δzf′+λz2f′′=2k+1∑n=2∞n-1ΦnanznB-Az+∑n=2∞B+1Ψn-A+1Φnanzn≤2k+1∑n=2∞ΦnanB-A-∑n=2∞B+1Ψn-A+1Φnan.
This expression is bounded above by1 if (10)∑n=2∞2k+1n-1Φn,δ,λ+B+1Ψn,δ,λ-A+1Φn,δ,λan≤B-A.Whenδ=0, we obtain the main result proved by Noor et at. [8] stated as follows.Corollary 3.
A functionf∈A given by (1) is in the class k-USTA,B, if it satisfies the following condition: (11)∑n=2∞2k+1n-1+B+1n-A+1an≤B-A.Whenδ=1, λ=0, then we have the following known result proved by Noor et at. [8] stated as follows.Corollary 4.
A functionf∈A given by (1) is in the class k-USTA,B, if it satisfies the following condition: (12)∑n=2∞n2k+1n-1+B+1n-A+1an≤B-A.Theorem 5.
A necessary and sufficient condition for any functionfz given by (2) to be in the class k-VUSTA,B,δ,λ is that(13)∑n=2∞2k+1n-1Φn+B+1Ψn-A+1Φnan≤B-A.Proof.
In view of Theorem2, we need only to prove the necessity. If f∈k-VUSTA,B,δ,λ and z is real, then, by relation (6), we have (14)B-A-∑n=2∞B-1Ψn-A-1Φnanzn-1B-A-∑n=2∞B+1Ψn-A+1Φnanzn-1≥2kn-1Φnanzn-1B-A-∑n=2∞B+1Ψn-A+1Φnanzn-1.Lettingz→1- along the real axis, we obtain the required inequality (13).Corollary 6.
Letfz∈k-VUSTA,B,δ,λ and be of the form (2). Then(15)an≤B-A2k+1n-1Φn+B+1Ψn-A+1Φn,n≥2.Next, we give the growth and distortion theorem for the classk-VUSTA,B,δ,λ.Theorem 7.
Let the functionfz given by (2) be in the class k-VUSTA,B,δ,λ. Then, for z<1, we have(16)fz≥r-B-A2k+1Φ2+B+1Ψ2-A+1Φ2r2,and(17)fz≤r+B-A2k+1Φ2+B+1Ψ2-A+1Φ2r2.Proof.
From Theorem5, for n≥2, we have (18)2k+1Φ2+B+1Ψ2-A+1Φ2∑n=2∞an≤2k+1n-1Φn+B+1Ψn-A+1Φn∑n=2∞an≤B-A.Forfz given by (2) and using the triangle inequality we have (19)fz≥z-∑n=2∞anzn≥r-r2∑n=2∞an≥r-B-A2k+1Φ2+B+1Ψ2-A+1Φ2r2,and(20)fz≤z+∑n=2∞anzn≤r+∑n=2∞anrnfz≤r+B-A2k+1Φ2+B+1Ψ2-A+1Φ2r2.Theorem 8.
Letfz∈k-VUSTA,B,δ,λ be of the form (1). Then, for z<1, we have(21)1-2B-A2k+1Φ2+B+1Ψ2-A+1Φ2r≤f′z≤1+2B-A2k+1Φ2+B+1Ψ2-A+1Φ2r.Proof.
By differentiating (2) and after some simplification we have(22)f′z≥1-∑n=2∞nanzn-1≥1-r∑n=2∞nan,
and(23)f′z≤1+∑n=2∞nanzn-1≤1+r∑n=2∞nan.
Asf∈k-VUSTA,B,δ,λ so, from Theorem 5, (24)2k+1Φ2+B+1Ψ2-A+1Φ22∑n=2∞nan≤2k+1n-1Φn+B+1Ψn-A+1Φn∑n=2∞nan≤B-A,
or equivalently(25)∑n=2∞nan≤2B-A2k+1Φ2+B+1Ψ2-A+1Φ2.
Using (25) in (22) and (23) yields required inequality (23).Theorem 9.
Letμν≥0, for ν=1, 2, …, l and ∑ν=1l‍μν≤1. If the functions Fνz=z-∑n=2∞‍an,νzn,an,ν≥0 are in the class k-VUSTA,B,δ,λ then(26)fz=z-∑n=2∞AnznAn=∑ν=1lμνan,ν,is in classk-VUSTA,B,δ,λ.Proof.
The proof follows immediately by using Theorem5.Theorem 10.
Letμn≥0, ∑n=1∞‍μn=1, f1z=z, and(27)fnz=z-∑n=2∞B-A2k+1n-1Φn+B+1Ψn-A+1Φnzn,n≥2.Thenfz is in the class k-VUSTA,B,δ,λ if and only if it can be expressed in the form(28)fz=∑n=1∞μnfnz,Proof.
Consider(29)fz=∑n=1∞μnfnz=z-∑n=2∞B-A2k+1n-1Φn+B+1Ψn-A+1Φnμnznn≥2.
Then, by Theorem5, we have(30)∑n=2∞2k+1n-1Φn+B+1Ψn-A+1ΦnB-A×B-A2k+1n-1Φn+B+1Ψn-A+1Φnμn=∑n=2∞μn=1-μ1<1,and hence fz∈k-VUSTA,B,δ,λ.
Conversely assume thatfz given by (2) is in the class k-VUSTA,B,δ,λ. Then (31)an≤B-A2k+1n-1Φn+B+1Ψn-A+1Φnn≥2.
By setting(32)μn=2k+1n-1Φn+B+1Ψn-A+1ΦnB-An≥2,
andμ1=1-∑n=2∞‍μn, we can see that fz can be expressed in the form (28), which completes the proof.Theorem 11.
Let the functionfz of the form (2) be in the class k-VUSTA,B,δ,λ. Then fz is close to convex of order ρ0≤ρ<1 in z<r1,(33)r1=infn≥21-ρ2k+1n-1Φn+B+1Ψn-A+1ΦnnB-A1/n-1.Proof.
It is well-known thatf∈K for z<r1 if (34)f′z-1≤1-ρforz<r1
(see [12]), where r1 is given by (33).
From (2), we have (35)f′z-1≤∑n=2∞nanzn-1.
Clearlyf′z-1≤1-ρ, if(36)∑n=2∞n1-ρzn-1≤1.
Asf∈k-VUSTA,B,δ,λ, from Theorem 5, inequality (36) will be true if (37)n1-ρzn-1≤2k+1n-1Φn+B+1Ψn-A+1ΦnB-A,
and this implies(38)z≤1-ρ2k+1n-1Φn+B+1Ψn-A+1ΦnnB-A1/n-1.Theorem 12.
Letfz of form (2) be in the class k-VUSTA,B,δ,λ. Then fz is starlike of order ρ0≤ρ<1 in z<r2,(39)r2=1-ρ2k+1n-1Φn+B+1Ψn-A+1Φnn-ρB-A1/n-1.Proof.
To provef∈STρ, for z<r2 it is sufficient to show (40)zf′zfz-1≤1-ρforz<r2
(see [12]). From (2) we have (41)zf′zfz-1≤∑n=2∞n-1anzn-11-∑n=2∞anzn-1.
Thuszf′z/fz-1≤1-ρ, if(42)∑n=2∞n-ρ1-ρanzn-1≤1.
Asf∈k-VUSTA,B,δ,λ, (42) must hold if (43)n-ρ1-ρzn-1≤2k+1n-1Φn+B+1Ψn-A+1ΦnB-A,
which gives usz<r2, where r2 is given by (39).Next we prove that the classk-VUSTA,B,δ,λ is closed under generalized Bernardi-Livingston operator defined as(44)Lcf=c+1zc∫0ztc-1ftdt,c>-1(see [17]).Theorem 13.
Iffz∈k-VUSTA,B,δ,λ then Lcf∈k-VUSTA,B,δ,λ.Proof.
Consider(45)Lcf=c+1zc∫0ztc-1ftdt,
using (2) and, after integration, we obtain(46)Lcf=z-∑n=2∞c+1c+nanzn.
Sincefz∈k-VUSTA,B,δ,λ, so an easy calculation leads to (47)∑n=2∞2k+1n-1Φn+B+1Ψn-A+1Φnc+1c+nan≤∑n=2∞2k+1n-1Φn+B+1Ψn-A+1Φnan≤B-A,
and, therefore,Lcf∈k-VUSTA,B,δ,λ.



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