# Majorization for A Subclass ofβ-Spiral Functions of Order α Involving a Generalized Linear Operator

Authors: Afaf A. Ali Abubaker; Maslina Darus; Daniel Breaz
Journal: Advances in Decision Sciences
 (2011)

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## Abstract

Motivated by Carlson-Shaffer linear operator, we define here a new generalized linear operator. Using this operator, we define a class of analytic functions in the unit diskU. For this class, a majorization problem of analytic functions is discussed.


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## Body

## 1. Introduction

LetA denote the class of functions f(z) of the form(1.1)f(z)=z+∑n=1∞an+1zn+1 which are analytic in the unit disk U={z∈ℂ:|z|<1}.Letf and g be analytic in U. Then, we say that function f is subordinate to g if there exists a Schwarz function ω(z), analytic in U with ω(0)=0 and |ω(z)|<1, such that f(z)=g(ω(z)), z∈U (see [1]). We denote this subordination by(1.2)f≺g(z∈U).Further,f is said to be quasi subordinate to g if there exists an analytic function φ(z) such that f(z)/φ(z) is analytic in U,(1.3)f(z)φ(z)≺g(z),(z∈U)
and |φ(z)|≤1. Note that the quasi subordination (1.3) is equivalent to(1.4)f(z)=φ(z)g(ω(z)),
where |φ(z)|≤1 and |ω(z)|≤|z|<1 (see [2]). If φ(z)=1, then (1.3) becomes (1.2).Let functionsf and g be analytic functions in U. If |f(z)|≤|g(z)|, then there exists a function φ analytic in U such that |φ(z)|≤1 in U, for which(1.5)f(z)=φ(z)g(z)(z∈U).
In this case, we say that f is majorized by g in U (see [3]), and we write(1.6)f(z)≪g(z)(z∈U).
If we take ω(z)=z in (1.4), then the quasi subordination (1.3) becomes the majorization (1.6).Also, letS denote the subclass of A consisting of all functions which are univalent in U.In [4], Robertson introduced star-like functions of order α on U.Definition 1.1.
Let0≤α<1 and f∈A; then, f is a star-like function of order α on U if and only if
(1.7)R{zf′(z)f(z)}>α(z∈U).
Let S*(α) denote the whole star-like functions of order α in U.Spaček [5] extended the class of S* and obtained the class of β-spiral-like functions. In the same article, the author gave an analytical characterization of spirallikeness of type β on U.Definition 1.2.
Let-π/2<β<π/2 and f∈A; then, f is β-spiral-like function on U if and only if
(1.8)R{eiβf′(z)f(z)}>0(z∈U).
We denote the whole β-spiral-like functions in U by Sβ*.Finally, Libera [6] introduced and studied the class of β-spiral-like functions of order α.Definition 1.3.
Let0≤α<1, -π/2<β<π/2 and f∈A; then, f is β-spiral function of order α if and only if
(1.9)R{eiβzf′(z)f(z)}>αcos⁡⁡β(z∈U).
We denote the whole β-spiral-like functions of order α in U by Sβ*(α).In particular, we consider the convolution with functionϕ(a,c) defined by (1.10)L(a,b)f(z)=z+∑n=1∞(a)n(c)nzn+1,
where a∈ℂ, b≠0,-1,-2,…, and (a)n is the Pochhammer symbol defined by (1.11)(a)n=Γ(a+n)Γ(a)={1,n=0,a(a+1)⋯(a+n-1),n={1,2,3,…}.
Function ϕ(a,c) is an incomplete beta-function related to the Gauss hypergeometric function by (1.12)ϕ(a,c;z)=z2F1(1,a;c;z).
It has an analytic continuation to the z-plane cut along the positive real line from 1 to ∞. We note that ϕ(a.1;z)=z/(1-z)a and ϕ(2,1;z) are the Koebe functions.Carlson and Shaffer [7] defined a convolution operator on A involving an incomplete beta-function as (1.13)L(a,b)f(z)=ϕ(a,c;z)*f(z)=z+∑n=1∞(a)n(c)nan+1zn+1.Definition 1.4.
Let functionF be given by
(1.14)F(m,l,λ)=∑n=0∞(1+l+λn1+l)mzn+1,
where ℓ,λ≥0 and m∈ℤ. The generalized linear operator L(m,ℓ,λ,a,c):A→A is given as
(1.15)L(m,l,λ,a,b)f(z)=z+∑n=1∞(1+l+λn1+l)m(a)n(c)nan+1zn+1.We note here some special cases.(1)
L(0,ℓ,λ,a,b)f(z)=L(a,b)f(z) is the Carlson and Shaffer operator [7].(2)
L(0,ℓ,λ,δ+1,1)f(z),δ∈ℕ0, is the Ruscheweyh derivative [8].(3)
L(m,0,λ,1,1)f(z),m∈ℕ0, is the Al-Oboudi operator [9].(4)
L(m,0,λ,a,b)f(z) is the linear operator introduced by Al-Refai and Darus [10].(5)
F(m,ℓ,λ),m∈ℕ0, is the generalized multiplier transformation which was introduced and studied by Cátáş [11].(6)
F(m,ℓ,1),m∈ℕ0, is the multiplier transformation which was introduced and studied by Cho and Srivastava [12] and Cho and Kim [13].Remark 1.5.
It follows from the above definition that(1.16)z(L(m,l,λ,a,c)f(z))′=aL(m,l,λ,a+1,c)f(z)-(a-1)L(m,l,λ,a,c)f(z)(z∈U).We introduce the classSβ*(m,ℓ,λ,a,c,α) as follows.Definition 1.6.
Leta∈ℂ, c≠0,-1,-2,…,ℓ, λ≥0, m∈ℤ, 0≤α<1, -π/2<β<π/2, and f∈A; then, one hasSβ*(m,ℓ,λ,a,c,ℓ,λ,α) if and only if
(1.17)R{eiβz(L(a,c)f(z))′L(a,c)f(z)}>αcos⁡⁡β.
Obviously, when a=c=1 and m=0 we obtain f∈Sβ*(α), when a=c=1 and m=β=0, we obtain that f(z) is a starl-like function of order α on U, and also when a=c=1 and m=α=0, we obtain that f(z) is spiral-like function of type β on U.Biernacki [14] in 1936 obtained the first results of majorization-subordination theory. He showed that, if g(z)∈S and f(z)≺g(z) in U, then f(z)≪g(z) in |z|≤(1/4). Goluzin [15] improved the result and Shah [16] obtained the complete solution for S by showing that f(z)≪g(z) in |z|≤(3-5)/2 and that the result is the best possible. A majorization problem for star-like functions has been given by MacGregor [3]. Also, majorization problem for star-like functions of complex order has recently been investigated by Altintaş et al. [17].The main object of this paper is to investigate the problem of majorization of the classSβ*(ℓ,λ,a,c,α) defined by a generalized linear operator.In order to prove our main theorem we need the following lemma.Lemma 1.7 (see [18]).
Letφ(z) be analytic in U satisfying |φ(z)|≤1 for z∈U. Then,
(1.18)|φ′(z)|≤1-|φ(z)|21-|z|2.


## 2. Main Results

Theorem 2.1.
Let functionf∈A and suppose that g∈Sβ*(m,ℓ,λ,a,c,α). If L(m,ℓ,λ,a,c)f is majorized by L(m,ℓ,λ,a,c)g in U, then
(2.1)|L(m,l,λ,a+1,c)f(z)|≤|L(m,l,λ,a+1,c)g(z)|(|z|≤r1),
where
(2.2)r1=r(m,l,λ,a,c,α,β)=2+|a|+|2(1-α)cos⁡⁡β-aeiβ|2|2(1-α)cos⁡⁡β-aeiβ|-Θ(a,α,β)2|2(1-α)cos⁡⁡β-aeiβ|,(2.3)Θ(a,α,β)=4+|a|2+|2(1-α)cos⁡⁡β-aeiβ|2+4|a|+4|2(1-α)cos⁡⁡β-aeiβ|-2|a||2(1-αcos⁡⁡β)-aeiβ|,
for a∈ℂ,c≠0,-1,-2,…,,ℓ,λ≥0,m∈ℤ,0≤α<1,-π/2<β<π/2, and |a|≥|2(1-α)cosβ-aeiβ|.Proof.
Sinceg∈Sβ*(m,ℓ,λ,a,c,α), we have
(2.4)eiβz(L(m,l,λ,a,c)g(z))′L(m,l,λ,a,c)g(z)=1+(1-2α)ω1-ωcos⁡⁡β+isin⁡⁡β,
where ω is analytic in U, with ω(0)=0 and
(2.5)|ω|≤|z|<1(z∈U).
By using (1.16) in (2.4), we get
(2.6)eiβ[aL(m,l,λ,a+1,c)g(z)-(a-1)L(m,l,λ,a,c)g(z)]L(m,l,λ,a,c)g(z)=1+(1-2α)ω1-ωcos⁡⁡β+isin⁡⁡β. Hence,
(2.7)L(m,l,λ,a+1,c)g(z)L(m,l,λ,a,c)g(z)=aeiβ+(2(1-α)cos⁡⁡β-aeiβ)ωaeiβ(1-ω),
which, in view of (2.5), immediately yields the inequality
(2.8)|L(m,l,λ,a,c)g(z)|≤|eiβ||a|(1+|z|)|a|-|2(1-α)cos⁡⁡β-aeiβ||z||L(m,l,λ,a+1,c)g(z)|.
Next, since L(m,ℓ,λ,a,c)f is majorized by L(m,ℓ,λ,a,c)g in U, from (1.5) we have
(2.9)z(L(m,l,λ,a,c)f(z))′=zφ′(z)L(m,l,λ,a,c)g(z)+zφ(z)(L(m,l,λ,a,c)g(z))′.
Also, by using (1.16) in (2.11), we get
(2.10)aL(m,l,λ,a+1,c)f(z)-(a-1)L(m,l,λ,a,c)f(z)=zφ′(z)L(m,l,λ,a,c)g(z)+φ(z)[aL(m,l,λ,a+1,c)g(z)-(a-1)L(m,l,λ,a,c)g(z)];
then, we have
(2.11)L(m,l,λ,a+1,c)f(z)=1azφ′(z)L(m,l,λ,a,c)g(z)+φ(z)L(m,l,λ,a+1,c)g(z).
Thus, by Lemma 1.7, since the Schwarz function ϕ satisfies the inequality in (1.18) and using (2.8) in (2.11), we get
(2.12)|L(m,l,λ,a+1,c)f(z)|≤(1-|φ(z)|2)|z|(1-|z|)(|a|-|2(1-α)cos⁡⁡β-aeiβ||z|)×|L(m,l,λ,a+1,c)g(z)|+|φ(z)||L(m,l,λ,a+1,c)g(z)|.
Hence,
(2.13)|L(m,l,λ,a+1,c)f(z)|≤(1-|φ(z)|2)|z|+(1-|z|)(|a|-|2(1-α)cos⁡⁡β-aeiβ||z|)|φ(z)|(1-|z|)(|a|-|2(1-α)cos⁡⁡β-aeiβ||z|)×|L(m,l,λ,a+1,c)g(z)|,
which, upon setting
(2.14)|z|=r,|φ(z)|=ρ(0≤ρ≤1)
yields
(2.15)|L(m,l,λ,a+1,c)f(z)|≤θ(ρ)(1-r)(|a|-|2(1-α)cos⁡⁡β-aeiβ|r)|L(m,l,λ,a+1,c)g(z)|,
where function θ(ρ) defined by
(2.16)θ(ρ)=(1-ρ2)r+(1-r)(|a|-|2(1-α)cos⁡β-aeiβ|r)ρ
takes on its maximum value at ρ=1 with
(2.17)r1=r(m,l,λ,a,c,α,β)=max⁡⁡{r∈[0,1]:ψ(r,ρ)≤1,∀ρ∈[0,1]},
where
(2.18)ψ(r,ρ)=θ(ρ)(1-r)(|a|-|2(1-α)cos⁡⁡β-aeiβ|r);
then, we have
(2.19)θ(ρ)(1-r)(|a|-|2(1-α)cos⁡⁡β-aeiβ|r)≤1.
A simple calculus in (2.19) is equivalent to
(2.20)-(1+ρ)r+(1-r)(|a|-|2(1-α)cos⁡⁡β-aeiβ|r)≥0,
while the inequality in (2.19) takes its minimum value at ρ=1, that is,
(2.21)|2(1-α)cos⁡⁡β-aeiβ|r2-(2|a|+|2(1-α)cos⁡⁡β-aeiβ|)r+|a|≥0,
for all r∈[0,r1], where r1=r(m,ℓ,λ,a,c,α,β) given in (2.2) holds true for |z|≤r(m,ℓ,λ,a,c,α,β), which proves the conclusion (2.1).Puttingm=α=β=0 in Theorem 2.1, we obtain the following result.Corollary 2.2.
Let functionf∈A and suppose that g∈S*(a,c). If L(a,c)f is majorized by L(a,c)g in U, then
(2.22)|L(a+1,c)f(z)|≤|L(a+1,c)g(z)|(|z|≤r2=r(a,c)),
where
(2.23)r(a,c)=3+|a|+|2-a|2|2-a|-4+|2-a|2-2|a||2-a|+4|a|+|a|22|2-a|.Further, puttinga=c=1 and m=0 in Theorem 2.1, we obtain the result of Altintaş et al. [17].Corollary 2.3.
Let functionf∈A and suppose that g∈S*((α-1)eiβ)=Sβ*(α), where 0≤α<1 and -π/2<β<π/2. If f is majorized by g in U, then
(2.24)|f′(z)|≤|g′(z)|(|z|≤r3=r(α,β)),
where
(2.25)r(α,β)=3+|2(α-1)eiβ-1|2|2(α-1)eiβ-1|-9+|2(α-1)eiβ-1|2+2|2(α-1)-1|2|2(α-1)eiβ-1|.Puttingβ=0 in Corollary 2.3, we obtain the result as follows.Corollary 2.4.
Let functionf∈A and suppose that g∈S*(α), where 0≤α<1. If f is majorized by g in U, then
(2.26)|f′(z)|≤|g′(z)|(|z|≤r4=r(α)),
where
(2.27)r(α)=3+|1-2α|2|1-2α|-9+|1-2α|2+2|2(α-1)-1|2|1-2α|.Also, puttingα=β=0 in Corollary 2.3, we obtain the result of MacGregor [3].Corollary 2.5.
Let functionf∈A and suppose that g∈S*(0). If f is majorized by g in U, then
(2.28)|f′(z)|≤|g′(z)|(|z|≤2-3).



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