# Coefficient Inequalities of Analytic Functions Related to Robertson Functions

Authors: Muhammad Arif; Mumtaz Ali
Journal: Journal of Mathematics
 (2013)

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## Abstract

We introduce and study a subclass of analytic functions related to  Robertson functions. Here we discuss the coefficient estimate for function in this class.


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## Body

## 1. Introduction

LetA be the class of functions f of the form
(1)f(z)=z+∑n=2∞‍anzn
which are analytic in the open unit disc E={z:|z|<1}. Also let S* and C denote the well-known classes of starlike and convex functions, respectively.For any two analytic functionsf given by (1) and g with
(2)g(z)=z+∑n=2∞‍bnzn,forz∈E,
the convolution (Hadamard product) is given by
(3)(f⋆g)(z)=z+∑n=2∞‍anbnzn,forz∈E.
Using the concept of convolution, Ruscheweyh [1] introduced a differential operator Dδ given by
(4)Dδf(z)=z(1-z)δ+1*f(z)=z+∑n=2∞‍φn(δ)anzn,(δ>-1)
with
(5)φn(δ)=(δ+1)n-1(n-1)!,
where (x)n is a Pochhammer symbol given as
(6)(x)n={1,n=0,x(x+1)(x+2)⋯(x+n-1),n∈ℕ.
It is obvious that D0f(z)=f(z), D1f(z)=zf′(z), and
(7)Dnf(z)=z(zn-1f(z))(n)n!,∀δ=n∈N0={0,1,2,…}.
The following identity can easily be established:
(8)(δ+1)Dδ+1f(z)=δDδf(z)+z(Dδf(z))′.
Now with the help of Ruscheweyh derivative, we define a class VDλ(α,β,b,δ) of analytic functions as follows.Definition 1.
Letf(z)∈A. Then, f(z)∈VDλ(α,β,b,δ), if and only if
(9)Re{eiλ(1-2b+2bDδ+1f(z)Dδf(z))}>α|2b(Dδ+1f(z)Dδf(z)-1)|+βcosλ,
where α≥0, 0≤β<1, δ>-1, λ is real with |λ|<π/2, and b∈ℂ∖{0}.
By giving specific values toα, β, λ, b, and δ in VDλ(α,β,b,δ), we obtain many important subclasses studied by various authors in earlier papers, see for details [2–5], and list some of them as follows: (i)
VDλ(0,0,2,0)≡Sλ* and VDλ(0,0,1,1)≡Kλ, studied by Spacek [6] and Robertson [7], respectively; for the advancement work, see [8, 9].(ii)
VD0(α,β,2,0)≡SD(α,β) and VD0(α,β,1,1)≡KD(α,β), studied by both Owa et al. and Shams et al. [10, 11].(iii)
VDλ(1,0,2,0)≡USP(λ), VDλ(1,0,1,1)≡UCSP(λ), introduced by Ravichandran et al. [12].(iv)
VD0(α,β,b,δ)≡VD(α,β,b,δ), considered by Latha [13].(v)
VD0(0,β,2,0)≡S*(β), VD0(0,β,1,1)≡C(β), the well-known classes of starlike and convex functions of order β.
From the above special cases, we note that this class provides a continuous passage from the class of starlike functions to the class of convex functions.
We will assume throughout our discussion, unless otherwise stated, thatα≥0, 0≤β<1, δ>-1, λ is real with |λ|<π/2, and b∈ℂ∖{0}.


## 2. Some Properties of the ClassVDλ(α,β,b,δ)

Theorem 2.
Iff(z)∈VDλ(α,β,b,δ) with 0≤α≤β, then
(10)f(z)∈VDλ(0,(β-α1-α),b,δ).Proof.
SinceRew≤|w| for any complex number w, f(z)∈VDλ(α,β,b,δ) implies that
(11)Re{eiλ(1-2b+2bDδ+1f(z)Dδf(z))}>α|2bDδ+1f(z)Dδf(z)-2b|+βcosλ≥αRe{eiλ(1-2b+2bDδ+1f(z)Dδf(z))}+(β-α)cosλ
which implies that
(12)Re{eiλ(1-2b+2bDδ+1f(z)Dδf(z))}>(β-α)cosλ(1-α),(z∈E).
And hence, we obtain the required result.
Putλ=0, b=2, and δ=0 in Theorem 2; we obtain the following result.Corollary 3 (see [10]).
Iff(z)∈SD(α,β) with 0≤α≤β, then
(13)f(z)∈S*(β-α1-α).
Set λ=0, b=1, and δ=1 in Theorem 2; one has the following result.Corollary 4 (see [10]).
Iff(z)∈KD(α,β) with 0≤α≤β, then
(14)f(z)∈K(β-α1-α).Theorem 5.
Iff(z)∈VDλ(α,β,b,δ), then
(15)|a2|≤|b||η||1-α|,(16)|an|≤(δ+1)|b||η|(n-1)|1-α|φn(δ)∏j=1n-2‍(1+(δ+1)|b||η|j|1-α|),n≥3,
where
(17)|η|=(1-β)2cos2λ+(1-α)2sin2λ.Proof.
We note that forf(z)∈VDλ(α,β,b,δ),
(18)Re{eiλ(1-2b+2bDδ+1f(z)Dδf(z))}>(β-α)(1-α)cosλ,z∈E.
Let us define the function p(z) by
(19)p(z)=((1-α)×[eiλ(1-2/b+(2/b)(Dδ+1f(z)/Dδf(z)))]-(β-α)cosλ)×((1-β)cosλ+i(1-α)sinλ)-1.
Then, p(z) is analytic in E with p(0)=1 and Rep(z)>0. Let
(20)p(z)=1+∑n=1∞pnzn.
Then, (19) can be written as
(21)1-2b+2bDδ+1f(z)Dδf(z)=1+(1-β)cosλ+i(1-α)sinλeiλ(1-α)∑n=1∞pnzn,
and using (8), we have
(22)2eiλ(1-α)(δ+1)(z(Dδf(z))′-(Dδf(z)))=bη(Dδf(z))(∑n=1∞‍pnzn)
which implies that
(23)eiλan=bη(δ+1)2(1-α)(n-1)φn(δ)×{pn-1+φ2(δ)a2pn-2+⋯+φn-1(δ)an-1p1},
where we have used (4) and (5). Now applying the coefficient estimates |pn|≤2 for Caratheodory function [14], we obtain
(24)|an|≤(δ+1)|b||η|(n-1)|1-α|φn(δ)×[1+φ2(δ)|a2|+⋯+φn-1(δ)|an-1|].
Forn=2,
(25)|a2|≤|b||η||1-α|
which proves (15).
Forn=3,
(26)|a3|≤(δ+1)|b||η|2|1-α|φ3(δ)[1+(δ+1)|b||η||1-α|].
Therefore, (16) holds for n=3. Assume that (16) is true for all n=3,4,…k and consider
(27)|ak+1|≤(δ+1)|b||η|(k+1-1)|1-α|φk+1(δ)×{{∏j=1k-21+(δ+1)|b||η|j|1-α|}(1+(δ+1)|b||η||1-α|)+(δ+1)|b||η|2|1-α|(1+(δ+1)|b||η||1-α|)+⋯+(δ+1)|b||η|(k-1)∣|1-α|∏j=1k-21+(δ+1)|b||η|j|1-α|}=(δ+1)|b||η|k|1-α|φk+1(δ)∏j=1k-1‍(1+(δ+1)|b||η|j|1-α|).
Thus, the result is true for n=k+1, and hence by induction, (16) holds for all n≥3.
If we setλ=0, b=2, and δ=0 in Theorem 5, we get the result proved in [10].Corollary 6.
Iff(z)∈SD(α,β), then
(28)|a2|≤2(1-β)|1-α|,|an|≤2(1-β)(n-1)|1-α|∏j=1n-2‍(1+2(1-β)j|1-α|),(n≥3).Remark 7.
If we takeα=0 in Corollary 6, we have
(29)|an|≤1(n-1)!∏j=2n‍(j-2β),(n≥2)
which was proved by Robertson [15].By settingλ=0, b=1, and δ=1 in Theorem 5, we obtain the result in [10].Corollary 8.
Iff(z)∈KD(α,β), then
(30)|a2|≤(1-β)|1-α|,|an|≤2(1-β)n(n-1)|1-α|∏j=1n-2(1+2(1-β)j|1-α|),(n≥3).Remark 9.
Lettingα=0 in Corollary 8, we have
(31)|an|≤1n!∏j=2n‍(j-2β),(n≥2)
given by Robertson [15].



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