# New Subclass ofK-Uniformly Univalent Analytic Functions with Negative Coefficients Defined by Multiplier Transformation

Authors: Ma’moun I. Y. Alharayzeh; Firas Ghanim
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

In this work, we shall derive a new subclass of univalent analytic functions denoted byHα,β,γ,k,λ,m in the open unit disk D=z∈ℂ:z<1 which is defined by multiplier transformation. Coefficient inequalities, growth and distortion theorem, extreme points, and radius of starlikeness and convexity of functions belonging to the subclass are obtained.


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## Body

## 1. Introduction and Definition

LetA denote the class of all analytic functions of the form
(1)fz=z+∑n=2∞anzn,aniscomplexnumber, defined on the open unit disk D=z∈ℂ:z<1 on the complex plane ℂ. Let S denote the subclass of A consisting of functions that are univalent in D. Further, let S∗α and Cα be the classes of functions, respectively, starlike of order α and convex of order α, for 0≤α<1. In particular, the classes S∗0=S∗ and C0=C are the familiar classes of starlike and convex functions in D, respectively.The functions of the form(2)fz=z−∑n=2∞anzn,an>0, defined on the open unit disk D=z∈ℂ:z<1, is called a functions with negative coefficients. Let T be the subclass of S, consisting of functions of the form (2). The class T was introduced and studied by Silverman [1]. In [1] Silverman investigated the subclasses of T denoted by ST∗α, and CTα, for 0≤α<1 that are, respectively, starlike of order α and convex of order α.LetMα,β,γ be the subclass of A consisting of functions fz which satisfy the inequality
(3)zf′z−fzαzf′z+1−γfz<β,where 0≤α≤1,0<β≤1, and 0≤γ<1 for all z∈D. This class of functions was studied by Darus [2].Cho and Srivastava [3] (see also [4]) introduced the operator Iλm as the following:Definition 1.
Forf∈A the multiplier transformation Iλm is defined by Iλm:A⟶A(4)Iλmfz=z+∑n=2∞n+λ1+λmanzn,z∈D,where −1<λ≤1 and m∈ℕ0=0,1,2.⋯.Special cases of this operator include the Uralegaddi and Somanatha operator in the caseλ=1 [5], and for λ=0, the operator Iλm reduces to well-known Salagean operator introduced by Salagean [6].By making use of the multiplier transformationIλm, the authors derive the new subclass Hα,β,γ,k,λ,m of functions as the following:Definition 2.
For0≤α≤1,0≤β<1,0≤γ<1,k≥0,−1<λ≤1, and m∈ℕ0=0,1,2,⋯, we let Hα,β,γ,k,λ,m consist of functions f∈T satisfying the condition
(5)RezIλmfz′−IλmfzαzIλmfz′+1−γIλmfz>kzIλmfz′−IλmfzαzIλmfz′+1−γIλmfz−1+β,where Iλmfz=z−∑n=2∞n+λ/1+λmanzn,z∈D and an>0.Our first result is the coefficient estimate for functionsf∈Hα,β,γ,k,λ,m, and the others include the growth and distortion theorem; further, we obtain the extreme points. Finally we determine the radius of starlikeness and convexity, for the function belonging to the class Hα,β,γ,k,λ,m.First of all, let us look at the coefficient estimates.


## 2. Coefficient Estimates

In this section, we shall obtain the coefficient estimates for the functionf belonging to the class Hα,β,γ,k,λ,m. Our first result is the following:Theorem 3.
Let0≤α≤1,0≤β<1,0≤γ<1,k≥0,−1<λ≤1, and m∈ℕ0=0,1,2,⋯. A function f given by (2) is in the class Hα,β,γ,k,λ,m if and only if
(6)∑n=2∞Tnα,β,γ,k,λ,man≤2+k−β1+α−γ,where
(7)Tnα,β,γ,k,λ,m=1+k2−γ−n1−α+1−β1+αn−γn+λ1+λm.Proof.
We havef∈Hα,β,γ,k,λ,m if and only if the condition (5) is satisfied.
Let(8)w=zIλmfz′−IλmfzαzIλmfz′+1−γIλmfz,subject to the condition that,
(9)Rew≥kw−1+βifandonlyifk+1w−1≤1−β.
Now(10)k+1w−1=k+1∑n=2∞1−nn+λ/1+λmanzn1+α−γz−∑n=2∞αn+1−γn+λ/1+λmanzn−1≤1−βis equivalent to
(11)k+1∑n=2∞1−nn+λ/1+λmanzn−11+α−γ−∑n=2∞αn+1−γn+λ/1+λmanzn−1−1≤1−β.
So(12)k+1∑n=2∞2−γ−n1−αn+λ/1+λmanzn−1−1+α−γ1+α−γ−∑n=2∞αn+1−γn+λ/1+λmanzn−1≤1−β.
The above inequality reduces to(13)k+1∑n=2∞2−γ−n1−αn+λ/1+λman−1+α−γ1+α−γ−∑n=2∞αn+1−γn+λ/1+λman≤1−β.
Then(14)k+1∑n=2∞2−γ−n1−αn+λ1+λman−1+α−γ≤1+α−γ1−β−1−β∑n=2∞αn+1−γn+λ1+λman.
Thus(15)∑n=2∞k+12−γ−n1−α+1−β∑n=2∞αn+1−γn+λ1+λman≤1+α−γ1−β+1+α−γk+1,which yield to (6).
Conversely suppose that (6) holds and we have to show that (12) holds. Here the inequality (6) is equivalent to (13). So it suffices to show that,
(16)∑n=2∞2−γ−n1−αn+λ/1+λmanzn−1−1+α−γ1+α−γ−∑n=2∞αn+1−γn+λ/1+λmanzn−1≤∑n=2∞2−γ−n1−αn+λ/1+λman−1+α−γ1+α−γ−∑n=2∞αn+1−γn+λ/1+λman.
Since(17)1+α−γ−∑n=2∞αn+1−γn+λ1+λmanzn−1≥1+α−γ−∑n=2∞αn+1−γn+λ1+λmanzn−1,we have
(18)1+α−γ−∑n=2∞αn+1−γn+λ1+λmanzn−1≥1+α−γ−∑n=2∞αn+1−γn+λ1+λman,wherez<1,and hence, we obtain (16).Theorem 4.
Let0≤α≤1,0≤β<1,0≤γ<1,k≥0,−1<λ≤1, and m∈ℕ0=0,1,2,⋯. If the function f given by (2) be in the class Hα,β,γ,k,λ,m, then
(19)an≤2+k−β1+α−γTnα,β,γ,k,λ,m,n=2,3,4,⋯,where Tnα,β,γ,k,λ,m is given by (7).
Equality holds for the functions given by,(20)fz=z−2+k−β1+α−γznTnα,β,γ,k,λ,m.Proof.
Sincef∈Hα,β,γ,k,λ,m, therefore, Theorem 3 holds.
Now(21)∑n=2∞Tnα,β,γ,k,λ,man≤2+k−β1+α−γ,we have,
(22)an≤2+k−β1+α−γTnα,β,γ,k,λ,m.
Clearly the function given by (20) satisfies (19), and, therefore, f given by (20) is in Hα,β,γ,k,λ,m for this function, and the result is clearly sharp.


## 3. Growth and Distortion Theorems for the SubclassHα,β,γ,k,λ,m

In this section, growth and distortion theorem will be considered, and the covering property for function in the classHα,β,γ,k,λ,m will also be given.Theorem 5.
Let0≤α≤1,0≤β<1,0≤γ<1,k≥0,−1<λ≤1, and m∈ℕ0=0,1,2,⋯. If the function f given by (2) is in the class Hα,β,γ,k,λ,m, then for 0<z=r<1, we have
(23)r−2+k−β1+α−γr2T2α,β,γ,k,λ,m≤fz≤r+2+k−β1+α−γr2T2α,β,γ,k,λ,m,where T2α,β,γ,k,λ,m is given by (7).
Equality holds for the function,(24)fz=z−2+k−β1+α−γz2T2α,β,γ,k,λ,m,z=±r,±ir.Proof.
We only prove the right hand side inequality in (23), since the other inequality can be justified using similar arguments.
Sincef∈Hα,β,γ,k,λ,m, then by Theorem 3, we have
(25)∑n=2∞Tnα,β,γ,k,λ,man≤2+k−β1+α−γ.
Now(26)T2α,β,γ,k,λ,m∑n=2∞an=∑n=2∞T2α,β,γ,k,λ,man≤∑n=2∞Tnα,β,γ,k,λ,man≤2+k−β1+α−γ.
And, therefore,(27)∑n=2∞an≤2+k−β1+α−γT2α,β,γ,k,λ,m.
Since(28)fz=z−∑n=2∞anzn,we have,
(29)fz=z−∑n=2∞anzn≤z+z2∑n=2∞anzn−2≤r+r2∑n=2∞an.
By aid of inequality (27), yields the right hand side inequality of (23).Theorem 6.
Let0≤α≤1,0≤β<1,0≤γ<1,k≥0,−1<λ≤1, and m∈ℕ0=0,1,2,⋯. If the function f given by (2) is in the class Hα,β,γ,k,λ,m for 0<z=r<1, then we have
(30)1−22+k−β1+α−γrT2α,β,γ,k,λ,m≤f′z≤1+22+k−β1+α−γrT2α,β,γ,k,λ,m,where T2α,β,γ,k,λ,m is given by (7).
Equality holds for the functionf given by
(31)fz=z−2+k−β1+α−γz2T2α,β,γ,k,λ,m,z=±r,±ir.Proof.
Sincef∈Hα,β,γ,k,λ,m, therefore, in Theorem 3, we have
(32)∑n=2∞Tnα,β,γ,k,λ,man≤2+k−β1+α−γ.
Now,(33)T2α,β,γ,k,λ,m∑n=2∞nan≤2∑n=2∞Tnα,β,γ,k,λ,man≤22+k−β1+α−γ.
Hence(34)∑n=2∞nan≤22+k−β1+α−γT2α,β,γ,k,λ,m.
Since(35)f′z=1−∑n=2∞nanzn−1.
Therefore, we have(36)1−z∑n=2∞nanzn−2≤f′z≤1+z∑n=2∞nanzn−2,wherez<1.
By using inequality (34), we get Theorem 6, and this completes the proof.Theorem 7.
Let0≤α≤1,0≤β<1,0≤γ<1,k≥0,−1<λ≤1, and m∈ℕ0=0,1,2,⋯. If the function f given by (2) is in the class Hα,β,γ,k,λ,m, then f is starlike of order δ, where
(37)δ=1−2+k−β1+α−γ−2+k−β1+α−γ+T2α,β,γ,k,λ,m.
The result is sharp with(38)fz=z−2+k−β1+α−γz2T2α,β,γ,k,λ,m,where T2α,β,γ,k,λ,m is given by (7).Proof.
It is sufficient to show that (6) implies
(39)∑n=2∞ann−δ≤1−δ.
That is,(40)n−δ1−δ≤Tnα,β,γ,k,λ,m2+k−β1+α−γ,n≥2.
The above inequality is equivalent to(41)δ≤1−2+k−β1+α−γn−1−2+k−β1+α−γ+Tnα,β,γ,k,λ,m=ψn,where n=2,3,4,⋯. And ψn≥ψ2, (40) holds true for any 0≤α≤1,0≤β<1,0≤γ<1,k≥0,−1<λ≤1, and m∈ℕ0=0,1,2,⋯. This completes the proof of Theorem 7.


## 4. Extreme Points of the SubclassHα,β,γ,k,λ,m

The main purpose of this section is to characterize the set of linear homeomorphisms of the extreme points of the closed convex hulls of the subclassHα,β,γ,k,λ,m, and the extreme points are given by the following theorem.Theorem 8.
Letf1z=z,
(42)fnz=z−2+k−β1+α−γznTnα,β,γ,k,λ,m,n=2,3,4,⋯,where Tnα,β,γ,k,λ,m is given by (7).
Thenf∈Hα,β,γ,k,λ,m if and only if it can be represented in the form
(43)fz=∑n=1∞ynfnz,where yn≥0and∑n=1∞yn=1.Proof.
Supposef can be expressed as in (43). Our goal is to show that f∈Hα,β,γ,k,λ,m .
By (43) we have
(44)fz=∑n=1∞ynz−2+k−β1+α−γznTnα,β,γ,k,λ,m.
Then(45)fz=z−∑n=2∞anzn=z−∑n=2∞2+k−β1+α−γynznTnα,β,γ,k,λ,m.
So that(46)an=2+k−β1+α−γynTnα,β,γ,k,λ,m,n≥2.
Now, we have(47)∑n=2∞yn=1−y1≤1.
Setting(48)∑n=2∞yn2+k−β1+α−γTnα,β,γ,k,λ,m×Tnα,β,γ,k,λ,m2+k−β1+α−γ=∑n=2∞yn=1−y1≤1.
It follows from Theorem3 that the function f∈Hα,β,γ,k,λ,m.
Conversely, it suffices to show that(49)an=2+k−β1+α−γynTnα,β,γ,k,λ,m.
Now we havef∈Hα,β,γ,k,λ,m, and then by Theorem 4, we have
(50)an≤2+k−β1+α−γTnα,β,γ,k,λ,m,n≥2.
That is,(51)Tnα,β,γ,k,λ,man2+k−β1+α−γ≤1,but yn≤1. Setting,
(52)yn=Tnα,β,γ,k,λ,man2+k−β1+α−γ,n≥2.
Yields the desired result. This completes the proof of the theorem.Corollary 9.
The extreme points of the classHα,β,γ,k,λ,m are the function
(53)f1z=z,fnz=z−2+k−β1+α−γznTnα,β,γ,k,λ,m,n=2,3,4,⋯,where Tnα,β,γ,k,λ,m is given by (7).
Finally, in this paper, we consider the radius of starlikeness and convexity for the classHα,β,γ,k,λ,m.


## 5. Radius of Starlikeness and Convexity

The radius of starlikeness and convexity for the classHα,β,γ,k,λ,m is given by the following theorems.Theorem 10.
If the functionf given by (2) is in the class Hα,β,γ,k,λ,m, then f is starlike of order δ0≤δ<1, in the disk z<R where
(54)R=infTnα,β,γ,k,λ,m2+k−β1+α−γ1−δn−δ1/n−1,n=2,3,4,⋯,where Tnα,β,γ,k,λ,m is given by (7).Proof.
Here (54) implies
(55)2+k−β1+α−γn−δzn−1≤Tnα,β,γ,k,λ,m1−δ.
It suffices to show that(56)zf′zfz−1≤1−δ,for z<R, we have
(57)zf′zfz−1≤∑n=2∞n−1anzn−11−∑n=2∞anzn−1.
By aid of (19), we have
(58)zf′f−1≤∑n=2∞n−12+k−β1+α−γzn−1/Tnα,β,γ,k,λ,m1−∑n=2∞2+k−β1+α−γzn−1/Tnα,β,γ,k,λ,m,the last expression is bounded above by 1−δ if,
(59)∑n=2∞n−12+k−β1+α−γzn−1Tnα,β,γ,k,λ,m≤1−∑n=2∞2+k−β1+α−γzn−1Tnα,β,γ,k,λ,m×1−δ,and it follows that
(60)zn−1≤Tnα,β,γ,k,λ,m2+k−β1+α−γ1−δn−δ,n≥2,which is equivalent to our condition (54) of the theorem.Theorem 11.
If the functionf given by (2) is in the class Hα,β,γ,k,λ,m, then f is convex of order ε0≤ε<1, in the disk z<w where
(61)w=infTnα,β,γ,k,λ,m2+k−β1+α−γ1−εnn−ε1/n−1,n=2,3,4,⋯,where Tnα,β,γ,k,λ,m is given by (7).Proof.
Here (61) implies
(62)2+k−β1+α−γnn−εzn−1≤Tnα,β,γ,k,λ,m1−ε.
It suffices to show that(63)zf′′zf′z≤1−ε,for z<w, we have
(64)zf′′zf′z≤∑n=2∞nn−1anzn−11−∑n=2∞nanzn−1.
Then by using the same technique in the proof of Theorem10, we can show that
(65)zf′′zf′z≤1−ε,forz≤w,with the aid of (19). Thus we have the assertion of Theorem 11.


## 6. Conclusion

The aim object of this article is to introduce a novel new subclass of univalent analytic functions on the open unit disc defined by multiplier transformation. We study their properties, begin by coefficients’ characterization. The functions in this class are acquired via this approach; for example, it can provide a number of fascinating features. The coefficient estimates, growth and distortion theorem, extreme points, starlikeness radius, and convexity of functions in the subclass are all introduced.We remark that several for a wider subclasses of univalent analytic functions can be introduced by using multiplier transformation and studied their coefficients’ characterization.



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