# Certain New Classes of Analytic Functions with Varying Arguments

Authors: R. M. El-Ashwah; M. K. Aouf; A. A. M. Hassan; A. H. Hassan
Journal: Journal of Complex Analysis
 (2013)

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## Abstract

We introduce certain new classesκ−VST(α,β) and κ−VUCV(α,β), which represent the κ uniformly starlike functions of order α and type β with varying arguments and the κ uniformly convex functions of order α and type β with varying arguments, respectively. Moreover, we give coefficients estimates, distortion theorems, and extreme points of these classes.


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## Body

## 1. Introduction

Let𝒜 denote the class of functions of the following form:
(1)f(z)=z+∑n=2∞‍anzn,
that are analytic and univalent in the open unit disc U={z∈ℂ:|z|<1}.Definition 1 (see [1]).
Letκ-ST(α,β) denote the subclass of 𝒜 consisting of functions f(z) of the form (1) and satisfy the following inequality:
(2)Re{zf′(z)f(z)-α}>κ|zf′(z)f(z)-β|,(0≤α<β≤1;κ(1-β)<(1-α);z∈U).
Also let κ-UCV(α,β) denote the subclass of 𝒜 consisting of functions f(z) of the form (1) and satisfy the following inequality:
(3)Re{1+zf′′(z)f′(z)-α}>κ|1+zf′′(z)f′(z)-β|,(0≤α<β≤1;κ(1-β)<(1-α);z∈U).
It follows from (2) and (3) that
(4)f∈κ-UCV(α,β)⟺zf′∈κ-ST(α,β).
The class κ-ST(α,β) denote the class of κ uniformly starlike functions of order α and type β and the class κ-UCV(α,β) denotes the class of κ uniformly convex functions of order α and type β.Specializing parametersβ,  α, and κ, we obtain the following subclasses studied by various authors:(i)
κ-ST(α,1)=SD(κ,α) and κ-UCV(α,1)=KD(κ,α) (see [2, 3]);(ii)
1-ST(α,1)=Sp(α) and 1-UCV(α,1)=UCV(α) (see [4]);(iii)
κ-ST(0,1)=κ-ST and κ-UCV(0,1)=κ-UCV (see [5, 6]);(iv)
1-ST(0,1)=Sp and 1-UCV(0,1)=UCV (see [4, 7–10]).Also we note that(5)1-ST(α,β)=US(α,β)={f(z)∈𝒜:Re{zf′(z)f(z)-α}>|zf′(z)f(z)-β|},1-UCV(α,β)=UC(α,β)={f(z)∈𝒜:Re{1+zf′′(z)f′(z)-α}>|1+zf′′(z)f′(z)-β|},
which are the uniformly starlike functions of order α and type β and uniformly convex functions of order α and type β, respectively.Definition 2 (see [11]).
A functionf(z) of the form (1) is said to be in the class V(θn) if f∈𝒜 and arg(an)=θn for all n≥2. If furthermore there exist a real number δ such that θn+(n-1)δ≡π(mod2π), then f(z) is said to be in the class V(θn,δ). The union of V(θn,δ) taken over all possible sequences {θn} and all possible real numbers δ is denoted by V.
Letκ-VST(α,β) denote the subclass of V consisting of functions f(z)∈κ-ST(α,β). Also Let κ-VUCV(α,β) denote the subclass of V consisting of functions f(z)∈κ-UCV(α,β).In this paper we obtain coefficient bounds for functions in the classesκ-VST(α,β) and κ-VUCV(α,β), respectively, further we obtain distortion bounds and the extreme points for functions in these classes.


## 2. Coefficient Estimates

Unless otherwise mentioned, we assume in the reminder of this paper that0≤α<β≤1, κ(1-β)<(1-α) and z∈U.We shall need the following lemmas.Lemma 3.
The sufficient condition forf(z)∈𝒜 given by (1) to be in the class κ-ST(α,β) is that
(6)∑n=2∞‍[κ(n-β)+(n-α)]|an|≤1-α-κ(1-β).Proof.
It suffices to show that inequality (2) holds true. Upon using the fact that
(7)Re{w}>κ|w-β|+α⟺Re{(1+κeiθ)w-βκeiθ}>α,
then inequality (2) may be written as
(8)Re{(1+κeiθ)zf′(z)f(z)-βκeiθ}>α,
or
(9)Re{A(z)B(z)}>α,
where A(z)=(1+κeiθ)zf′(z)-βκeiθf(z) and B(z)=f(z), then condition (2) or (9) is equivalent to
(10)|A(z)+(1-α)B(z)|-|A(z)-(1+α)B(z)|≥0.
We note that
(11)|A(z)+(1-α)B(z)|=|[(1-β)κeiθ+2-α]z∑n=2∞‍-∑n=2∞‍[(β-n)κeiθ+α-n-1]anzn|≥[-κ(1-β)+2-α]|z|-∑n=2∞‍[(n-β)κ+n-α+1]|an||z|n,(12)|A(z)-(1+α)B(z)|=|[(1-β)κeiθ-α]z∑n=2∞‍+∑n=2∞‍[(n-β)κeiθ+n-α-1]anzn|≤[κ(1-β)+α]|z|+∑n=2∞‍[(n-β)κ+n-α-1]|an||z|n.
Using (11) and (12), then we can obtain the following inequality:
(13)|A(z)+(1-α)B(z)|-|A(z)-(1+α)B(z)|≥2[(1-α)-κ(1-β)]|z|-2∑n=2∞‍[(n-β)κ+n-α]|an||z|n.
The expression |A(z)+(1-α)B(z)|-|A(z)-(1+α)B(z)| is bounded below by 0 if
(14)2[(1-α)-(1-β)κ]|z|-2∑n=2∞‍[(n-β)κ+n-α]|an||z|n>0,
or
(15)∑n=2∞‍[κ(n-β)+(n-α)]|an|<1-α-κ(1-β).
Hence the proof of Lemma 3 is completed.By using (4) and (6) we can obtain the following lemma.Lemma 4.
A functionf(z) of the form (1) is in the class κ-UCV(α,β) if
(16)∑n=2∞‍n[κ(n-β)+(n-α)]|an|<1-α-κ(1-β).Remark 5.
Puttingβ=1 in Lemmas 3 and 4, we obtain the results obtained by Shams et al. [3, Theorems 2.1, 2.2, resp.].In the following theorems, we show that the conditions (6) and (16) are also necessary for functions f(z)∈κ-VST(α,β) and κ-VUCV(α,β), respectively.Theorem 6.
Letf(z) be of the form (1), then f(z)∈κ-VST(α,β) if and only if
(17)∑n=2∞‍[κ(n-β)+(n-α)]|an|<1-α-κ(1-β).Proof.
In view of Lemma3, we need only to show that function f(z)∈κ-VST(α,β) satisfies the coefficient inequality (17). If f(z)∈κ-VST(α,β), then from (2), we have
(18)Re{zf′(z)f(z)-α}>κ|zf′(z)f(z)-β|,
thus we have
(19)Re{(1-α)+∑n=2∞(n-α)anzn-11+∑n=2∞anzn-1}>κ|(1-β)+∑n=2∞(n-β)anzn-11+∑n=2∞anzn-1|.
Since f(z)∈V, f(z)  lies in the class V(θn,δ) for some sequence {θn} and a real number δ such that
(20)θn+(n-1)δ≡π(mod2π).
Set z=reiδ in (19), then we obtain
(21)(1-α)-∑n=2∞(n-α)|an|rn-11-∑n=2∞anrn-1>κ[(1-β)+∑n=2∞(n-β)|an|rn-11-∑n=2∞|an|rn-1].
Letting r→1, then we have
(22)∑n=2∞‍[κ(n-β)+(n-α)]|an|<1-α-κ(1-β).
Hence the proof of Theorem 6 is completed.Corollary 7.
Iff(z)∈κ-VST(α,β), then
(23)|an|≤1-α-κ(1-β)κ(n-β)+(n-α)(n≥2).
The equality holds for the function
(24)f(z)=z+1-α-κ(1-β)κ(n-β)+(n-α)eiθnzn(n≥2;z∈U).Using the same technique used in Theorem6 we get the following theoremTheorem 8.
Letf(z) of the form (1), then f(z)∈κ-VUCV(α,β) if and only if
(25)∑n=2∞‍n[κ(n-β)+(n-α)]|an|<1-α-κ(1-β).Corollary 9.
Iff(z)∈κ-VUCV(α,β), then
(26)|an|≤1-α-κ(1-β)n[κ(n-β)+(n-α)](n≥2).
The equality holds for the function
(27)f(z)=z+1-α-κ(1-β)n[κ(n-β)+(n-α)]eiθnzn(n≥2;z∈U).


## 3. Distortion Theorems

Theorem 10.
Let the functionf(z) defined by (1) be in the class κ-VST(α,β). Then
(28)|z|-1-α-κ(1-β)κ(2-β)+(2-α)|z|2≤|f(z)|≤|z|+1-α-κ(1-β)κ(2-β)+(2-α)|z|2.
The result is sharp.Proof .
We employ the same technique as used by Silverman [11]. In view of Theorem 6, since
(29)Φ(n)=κ(n-β)+(n-α)
is an increasing function of n(n≥2), we have
(30)Φ(2)∑n=2∞‍|an|≤∑n=2∞‍Φ(n)|an|≤1-α-κ(1-β),
that is,
(31)∑n=2∞‍|an|≤1-α-κ(1-β)Φ(2).
Thus we have
(32)|f(z)|≤|z|+|z|2∑n=2∞‍|an|,|f(z)|≤|z|+1-α-κ(1-β)κ(2-β)+(2-α)|z|2.
Similarly, we get
(33)|f(z)|≥|z|-∑n=2∞‍|an||z|n≥|z|-|z|2∑n=2∞‍|an|≥|z|-1-α-κ(1-β)κ(2-β)+(2-α)|z|2.
This completes the proof of Theorem 10. Finally the result is sharp for the following function:
(34)f(z)=z+1-α-κ(1-β)κ(2-β)+(2-α)eiθ2z2,
at z=±|z|e-iθ2.Corollary 11.
Under the hypotheses of Theorem8, f(z) is included in a disc with center at the origin and radius r1 given by
(35)r1=1+1-α-κ(1-β)κ(2-β)+(2-α).Theorem 12.
Let the functionf(z) defined by (1) be in the class κ-VST(α,β). Then
(36)1-2[1-α-κ(1-β)]κ(2-β)+(2-α)|z|≤|f′(z)|≤1+2[1-α-κ(1-β)]κ(2-β)+(2-α)|z|.
The result is sharp.Proof.
SimilarlyΦ(n)/n is an increasing function of n(n≥2), in view of Theorem 6, we have
(37)Φ(2)2∑n=2∞‍n|an|≤∑n=2∞‍Φ(n)|an|≤1-α-κ(1-β),
that is,
(38)∑n=2∞‍n|an|≤2[1-α-κ(1-β)]Φ(2).
Thus we have
(39)|f′(z)|≤1+|z|∑n=2∞‍n|an|≤1+2[1-α-κ(1-β)]κ(2-β)+(2-α)|z|.
Similarly
(40)|f′(z)|≥1-|z|∑n=2∞‍n|an|≥1-2[1-α-κ(1-β)]κ(2-β)+(2-α)|z|.
Finally, we can see that the assertions of Theorem 12 are sharp for the function f(z) defined by (34). This completes the proof of Theorem 12.Corollary 13.
Under the hypotheses of Theorem12, f′(z) is included in a disc with center at the origin and radius r2 given by
(41)r2=1+2[1-α-κ(1-β)]κ(2-β)+(2-α).Using the same technique used in Theorems10 and 12, we get the following theorems.Theorem 14.
Let the functionf(z) defined by (1) be in the class κ-VUCV(α,β). Then
(42)|z|-1-α-κ(1-β)2[κ(2-β)+(2-α)]|z|2≤|f(z)|≤|z|+1-α-κ(1-β)2[κ(2-β)+(2-α)]|z|2.
The result is sharp for the following function:
(43)f(z)=z+1-α-κ(1-β)2[κ(2-β)+(2-α)]eiθ2z2.Theorem 15.
Let the functionf(z) defined by (1) be in the class κ-VUCV(α,β). Then
(44)1-1-α-κ(1-β)κ(2-β)+(2-α)|z|≤|f′(z)|≤1+1-α-κ(1-β)κ(2-β)+(2-α)|z|.
The result is sharp for the function given by (43).


## 4. Extreme Points

Theorem 16.
Let the functionf(z) defined by (1) be in the class κ-VST(α,β), with argan=θn, where θn+(n-1)δ≡π(mod2π). Define
(45)f1(z)=z,fn(z)=z+1-α-κ(1-β)κ(n-β)+(n-α)eiθnzn(n≥2;z∈U).
Then f(z)∈κ-VST(α,β) if and only if f(z) can be expressed in the form f(z)=∑n=1∞μnfn(z), where μn≥0 and ∑n=1∞μn=1.Proof.
Iff(z)=∑n=1∞μnfn(z) with μn≥0 and ∑n=1∞μn=1, then
(46)f(z)=∑n=1∞‍μnfn(z)=μ1f1(z)+∑n=2∞‍μnfn(z)=(1-∑n=2∞‍μn)z+∑n=2∞‍[μn(z+1-α-κ(1-β)κ(n-β)+(n-α)eiθnzn)]=z+∑n=2∞‍1-α-κ(1-β)κ(n-β)+(n-α)μneiθnzn.
But according to (17), we can see that
(47)∑n=2∞‍[κ(n-β)+(n-α)]|1-α-κ(1-β)κ(n-β)+(n-α)μneiθn|=∑n=2∞‍[κ(n-β)+(n-α)]1-α-κ(1-β)κ(n-β)+(n-α)μn=∑n=2∞‍[1-α-κ(1-β)]μn=(1-μ1)[1-α-κ(1-β)]≤[1-α-κ(1-β)].
Then f(z) satisfies (17), hence f(z)∈κ-VST(α,β).
Conversely, let the functionf(z) defined by (1) be in the class κ-VST(α,β), and define
(48)μn=[κ(n-β)+(n-α)]1-α-κ(1-β)|an|,n≥2,μ1=1-∑n=2∞‍μn.
From Theorem 6, ∑n=2∞μn≤1 and so μ1≥0. Since μnfn(z)=μnz+anzn, then
(49)∑n=1∞‍μnfn(z)=z+∑n=2∞‍anzn=f(z).
This completes the proof of Theorem 16.Finally using the same technique used in Theorem16 we get the following theorem.Theorem 17.
Let the functionf(z) defined by (1) be in the class κ-VUCV(α,β), with argan=θn, where θn+(n-1)δ≡π (mod2π). Define
(50)f1(z)=z,fn(z)=z+1-α-κ(1-β)n[κ(n-β)+(n-α)]eiθnzn(n≥2;z∈U).
Then f(z)∈κ-VUCV(α,β) if and only if f(z) can be expressed in the form f(z)=∑n=1∞μnfn(z), where μn≥0 and ∑n=1∞μn=1.



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