# Some Properties for Certain Class of Analytic Functions with Varying Arguments

Authors: M. K. Aouf; A. O. Mostafa; A. Shamandy; E. A. Adwan
Journal: International Journal of Analysis
 (2013)

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## Abstract

We introduce a new classV(k,λ,A,B) of analytic functions with varying arguments in the open unit disc U={z∈ℂ:|z|<1} defined by the Salagean operator. The object of the present paper is to determine coefficient estimates, extreme points, and distortion theorems for functions f(z) belonging to the class V(k,λ,A,B).


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## Body

## 1. Introduction

Let𝒜 denote the class of functions of the form
(1)f(z)=z+∑n=2∞‍anzn,
which are analytic in the open unit disc U={z∈ℂ:|z|<1}. If f and g  are analytic functions in U, we say that f is subordinate to g, written  f≺g if there exists a Schwarz function w, which (by definition) is  analytic in U with w(0)=0 and |w(z)|<1for all z∈U, such that f(z)=g(w(z)),z∈U. Furthermore, if the function g is univalent in U, then we  have the following equivalence (cf, e.g., [1, 2]):
(2)f(z)≺g(z)⟺f(0)=g(0),f(U)⊂g(U).
For function f(z)∈𝒜, Salagean [3] defined the differential operator Dk as follows:
(3)D0f(z)=f(z),D1f(z)=Df(z)=zf′(z),Dkf(z)=D(Dk-1f(z)),(k∈ℕ0=ℕ∪{0},ℕ={1,2,…}).
For λ≥0,-1≤A<B≤1,0<B≤1,k∈ℕ0, and for all z∈U, let  S(k,λ,A,B) denotes the subclass of 𝒜 consisting of functions f(z) of the form (1) and satisfying the analytic criterion:
(4)(1-λ)Dkf(z)z+λ(Dkf(z))′≺1+Az1+Bz.
It is noticed that for suitable choices of k,λ,A, and B, we obtain the following subclasses studied by various authors:(1)
S(k,λ,A,B)=Gk(λ,1,A,B)=Gk(λ,A,B)(λ≥0,-1≤A<B≤1,0<B≤1,k∈ℕ0) (see Sivasubramanian et al. [4], with b=1);(2)
S(k,λ,2α-1,1)=Sℑ(k,λ,α)(k∈ℕ0,0≤α<1,λ≥0) (see Aouf et al. [5], with b=1);(3)
S(0,λ,2α-1,1)=B(λ,α)(0≤α<1,0≤λ≤1) (see Chunyi and Owa [6]);(4)
S(0,0,2α-1,1)=B(α)(0≤α<1) (see Chen [7, 8] and Goel [9]);(5)
S(1,0,2α-1,1)=C(α)(0≤α<1) (see Srivastava and Owa [10]).Also we note the following.(1)
Puttingk=0 in (4), the class S(0,λ,A,B) reduces to the class S(λ,A,B)={(1-λ)(f(z)/z)+λf′(z)≺((1+Az)/(1+Bz)),λ≥0,-1≤A<B≤1,0<B≤1}.Definition 1 (see [11]).
A functionf(z) defined by (1) is said to be in the class V(θn) if f(z)∈𝒜 and arg(an)=θn for all n≥2. If, furthermore, there exists a  real number β such that
(5)θn+(n-1)β≡π(mod2π),
then f(z) is said to be in the class V(θn;β). The union of V(θn;β) taken over all possible sequences {θn} and all possible real numbers β is denoted by V.
LetV(k,λ,A,B) denote the subclass of V consisting of functions f(z) in S(k,λ,A,B).
We note the following:(1)
V(0,λ,2α-1,1)=Vλ(α)(0≤α<1,λ≥0) (see Aouf et al. [12]);(2)
V(0,0,2α-1,1)=Bα(0≤α<1) (see Srivastava and Owa [10]);(3)
V(1,0,2α-1,1)=Cα(0≤α<1) (see Srivastava and Owa [10]).Also we note that(i)
V(k,λ,2α-1,1)=Vℑ(k,λ,α)(k∈ℕ0,0≤α<1,λ≥0) denotes the subclass of V, consisting of functions f(z) belonging to the class Sℑ(k,λ,α);(ii)
V(0,λ,A,B)=V(λ,A,B)(λ≥0,-1≤A<B≤1,0<B≤1) denotes the subclass of V, consisting of functions f(z) belonging to the class S(λ,A,B).


## 2. Coefficient Estimates

Unless otherwise mentioned, we will assume in the reminder of this paper thatλ≥0,-1≤A<B≤1,0<B≤1 and k∈ℕ0.Theorem 2.
Let the functionf(z)∈𝒜 be of the form (1), if
(6)∑n=2∞nk[1+λ(n-1)](1+B)|an|≤(B-A),
then f(z)∈S(k,λ,A,B).Proof.
A functionf(z) of the form (1) belongs to the class S(k,λ,A,B) if and only if there exists a function w,|w(z)|≤|z|, such that
(7)(1-λ)Dkf(z)z+λ(Dkf(z))′=1+Aw(z)1+Bw(z),
or, equivalently,
(8)|(1-λ)(Dkf(z)/z)+λ(Dkf(z))′-1B[(1-λ)(Dkf(z)/z)+λ(Dkf(z))′]-A|<1.
Thus, it is sufficient to prove that
(9)|(1-λ)Dkf(z)z+λ(Dkf(z))′-1|-|B[(1-λ)Dkf(z)z+λ(Dkf(z))′]-A|<0.
Indeed, letting |z|=r(0≤r<1) we have
(10)|(1-λ)Dkf(z)z+λ(Dkf(z))′-1|-|B[(1-λ)Dkf(z)z+λ(Dkf(z))′]-A|=|∑n=2∞nk[1+λ(n-1)]anzn-1|-|(B-A)+∑n=2∞Bnk[1+λ(n-1)]anzn-1|≤(∑n=2∞nk[1+λ(n-1)]|an|rn-1-(B-A)+∑n=2∞Bnk[1+λ(n-1)]|an|rn-1∑n=2∞nk[1+λ(n-1)]|an|rn-1)<∑n=2∞nk[1+λ(n-1)](1+B)|an|-(B-A).
In view of (6), the last inequality is less than zero; hence f(z)∈S(k,λ,A,B).  This completes the proof of Theorem 2.Theorem 3.
Let the functionf(z)∈𝒜 be of the form (1), then f(z) is in the class V(k,λ,A,B) if  and only if
(11)∑n=2∞nk[1+λ(n-1)](1+B)|an|≤(B-A).Proof.
In view of Theorem2, we need only to show that each function f(z) from the class V(k,λ,A,B) satisfies the coefficient inequality (6). Let f(z)∈V(k,λ,A,B). Then, by (8) and (1), we have
(12)|∑n=2∞nk[1+λ(n-1)]anzn-1(B-A)+∑n=2∞Bnk[1+λ(n-1)]anzn-1|<1.
Since f(z)∈V, f(z) lies in the class V(θn,β) for some sequence {θn} and a real number β such that θn+(n-1)β≡π(mod2π)(n≥2), set z=reiβ in the above inequality and since Re{w(z)}<|w(z)|<1, then
(13)∑n=2∞nk[1+λ(n-1)]|an|rn-1(B-A)-∑n=2∞Bnk[1+λ(n-1)]|an|rn-1<1.
It is clear that the denominator of the left-hand said cannot vanish for r∈[0,1). Moreover, it is positive for r=0 and in consequence for r∈[0,1). Thus,  we have
(14)∑n=2∞nk[1+λ(n-1)](1+B)|an|rn-1≤(B-A),
which, upon lettingr→1-, readily yields the assertion (11). This completes the proof of Theorem 3.Corollary 4.
Let the functionf(z)∈𝒜 defined by (1) be in the classV(k,λ,A,B), then
(15)|an|≤(B-A)nk[1+λ(n-1)](1+B)(n≥2).
The result is sharp for the function
(16)f(z)=z+(B-A)nk[1+λ(n-1)](1+B)eiθnzn(n≥2).


## 3. Distortion Theorems

Theorem 5.
Let the functionf(z)∈𝒜 defined by (1) be in the classV(k,λ,A,B).Then
(17)|z|-(B-A)2k(1+B)(1+λ)|z|2≤|f(z)|≤|z|+(B-A)2k(1+B)(1+λ)|z|2.
The result is sharp.Proof.
Since(18)Ψ(n)=nk[1+λ(n-1)](1+B)
is an increasing function of n(n≥2), from Theorem 2, we have
(19)2k(1+B)(1+λ)∑n=2∞|an|≤∑n=2∞nk[1+λ(n-1)](1+B)|an|≤(B-A),
that is,
(20)∑n=2∞|an|≤(B-A)2k(1+B)(1+λ).
Thus,
(21)|f(z)|=|z+∑n=2∞anzn|≤|z|+∑n=2∞|an||z|2≤|z|+(B-A)2k(1+B)(1+λ)|z|2.
Similarly, we get
(22)|f(z)|≥|z|-∑n=2∞|an||zn|≥|z|-∑n=2∞|an||z|2≥|z|-(B-A)2k(1+B)(1+λ)|z|2.
Finally, the result is sharp for the function(23)f(z)=z+(B-A)2k(1+B)(1+λ)eiθ2z2
at z=±|z|e-iθ2.
This completes the proof of Theorem5.Corollary 6.
Under the hypotheses of Theorem5, f(z) is included in a disc with center at the origin and radius r1 given by
(24)r1=1+(B-A)2k(1+B)(1+λ).Theorem 7.
Let the functionf(z)∈𝒜 defined by (1) belong to the classV(k,λ,A,B)(k≥1).Then
(25)1-(B-A)2k-1(1+B)(1+λ)|z|≤|f′(z)|≤1+(B-A)2k-1(1+B)(1+λ)|z|.
The result is sharp for the function f(z) given by (23) at z=±|z|e-iθ2.Proof.
Since{nΨ(n)}, where Ψ(n) given by (18) is increasing function of n(n≥2), in view of Theorem 3, we have
(26)2k(1+B)(1+λ)2∑n=2∞n|an|≤∑n=2∞nk[1+λ(n-1)](1+B)|an|≤(B-A),
that is,
(27)∑n=2∞n|an|≤(B-A)2k-1(1+B)(1+λ).
Thus,
(28)|f′(z)|=|1+∑n=2∞nanzn-1|≤1+∑n=2∞n|an||z|≤1+(B-A)2k-1(1+B)(1+λ)|z|.
Similarly, we get
(29)|f′(z)|≥1-∑n=2∞n|an||zn-1|≥1-∑n=2∞n|an||z|≥1-(B-A)2k-1(1+B)(1+λ)|z|.
Finally, the result is sharp for the function f(z) given by (23). This completes the proof of Theorem 7.Corollary 8.
Let the functionf(z)∈𝒜 defined by (1) be in the class V(k,λ,A,B). Then f′(z) is included in a disc with center at the origin and radius r2 given by
(30)r2=1+(B-A)2k-1(1+B)(1+λ).


## 4. Extreme Points

Theorem 9.
Let the functionf(z)∈𝒜 defined by (1) be in the class V(k,λ,A,B), with arg(an)=θn where [θn+(n-1)β]≡π(mod2π). Define f1(z)=z and
(31)fn(z)=z+(B-A)nk[1+λ(n-1)](1+B)eiθnzn.
Then f(z)is in the class V(k,λ,A,B) if and only if it can be expressed in the form
(32)f(z)=∑n=1∞μnfn(z),whereμn≥0(n≥1)and ∑n=1∞μn=1.Proof.
Iff(z)=∑n=1∞μnfn(z) with ∑n=1∞μn=1 and μn≥0, then
(33)∑n=2∞nk[1+λ(n-1)](1+B)(B-A)nk[1+λ(n-1)](1+B)μn=∑n=2∞(B-A)μn=(B-A)(1-μ1)≤(B-A).
Hence, f(z)∈V(k,λ,A,B).
Conversely, let the functionf(z) defined by (1) belong to the class V(k,λ,A,B), and define
(34)μn=nk[1+λ(n-1)](1+B)(B-A)|an|,μ1=1-∑n=2∞μn.
From Theorem 3, ∑n=1∞μn≤1 and so μn≥0. Since μnfn(z)=μnz+anzn, then
(35)∑n=1∞μnfn(z)=z+∑n=2∞anzn=f(z).
This completes the proof of Theorem 9.Remark 10.
Specializingk,λ,A, and B in the above results, we obtain the corresponding results for the corresponding classes Vℑ(k,λ,α) and V(λ,A,B) defined in Section 1.



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