# Fekete-Szegő Inequality for a Subclass ofp-Valent Analytic Functions

Authors: Mohsan Raza; Muhammad Arif; Maslina Darus
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The main object of this paper is to study Fekete-Szegő problem for the class ofp-valent functions. Fekete-Szegő inequality of several classes is obtained as special cases from our results. Applications of the results are also obtained on the class defined by convolution.


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## Body

## 1. Introduction and Preliminaries

LetAp denote the class of functions f(z) of the form
(1)f(z)=zp+∑n=p+1∞‍anzn,(p∈ℕ={1,2,3,…}),
which are analytic in the open unit disk E. Also, A1=A, the usual class of analytic functions defined in the open unit disk E={z:|z|<1}. Let f(z) and g(z) be analytic in E. We say that the function f is subordinate to the function g and write f(z)≺g(z), if and only if there exists Schwarz function w, analytic in E such that w(0)=0, |w(z)|<1 for z∈E, and f(z)=g(w(z)). In particular, if g is univalent in E, then we have the following equivalence:
(2)f(z)≺g(z)⟺f(0)=g(0),f(E)⊂g(E).For any two analytic functionsf(z) of the form (1) and g(z) with
(3)g(z)=zp+∑n=p+1∞‍bnzn,z∈E,
the convolution (Hadamard product) is given by
(4)(f*g)(z)=zp+∑n=p+1∞‍anbnzn,z∈E.
Let ϕ(z) be an analytic function with positive real part on E with ϕ(0)=1, ϕ′(0)>0, which maps the unit disk E onto a region starlike with respect to 1 which is symmetric with respect to the real axis. Denote by Sp*(ϕ) the class of functions f analytic in E for which
(5)zf′(z)pf(z)≺ϕ(z),z∈E.
The classSp*(ϕ) was defined and studied by Ali et al. [1]. They obtained the Fekete-Szegő inequality for functions in the class Sp*(ϕ). The class S1*(ϕ) coincides with the class S*(ϕ) discussed by Ma and Minda [2]. Owa [3] introduced a subclass of p-valently Bazilevic functions Hp(A,B,α,β). A function f∈Ap is said to be in the class Hp(A,B,α,β) if and only if
(6)(1-β)(f(z)zp)α+βzf′(z)pf(z)(f(z)zp)α≺1+Az1+Bz,z∈E,
where -1≤B<A≤1, α≥0, and 0≤β≤1. We now define the following subclass of analytic functions.Definition 1.
Letϕ(z) be a univalent starlike function with respect to 1 which maps the unit disk E onto a region in the right half-plane which is symmetric with respect to the real axis with ϕ(0)=1 and ϕ′(0)>0. A function f∈Ap is in the class Vp,b,α,β(ϕ) if
(7)1-2b+2b{zf′(z)pf(z)(f(z)zp)α(1-β)(f(z)zp)α1-2b+2b+βzf′(z)pf(z)(f(z)zp)α}≺ϕ(z),
where 0≤β≤1, α≥0, and b>0.Definition 2.
A functionf∈Ap is in the class Vp,b,α,β,g(ϕ) if
(8)1-2b+2b{βz(f*g)′(z)p(f*g)(z)((f*g)(z)zp)α(1-β)((f*g)(z)zp)α1-2b+2b+βz(f*g)′(z)p(f*g)(z)((f*g)(z)zp)α}≺ϕ(z),
where 0≤β≤1, α≥0, and b>0.In other words, a function f∈Apis in the class Vp,b,α,β,g(ϕ) if (f*g)(z)∈Vp,b,α,β(ϕ).
We have the following special cases.(i)
Vp,2,1,1(ϕ) coincides with the class Sp*(ϕ) introduced and studied by Ali et al. [1].(ii)
Forp=1, b=2, and β=1, we have the class Bα(ϕ) introduced and studied by Ravichandran et al. [4].(iii)
Forb=2 and ϕ(z)=(1+Az)/(1+Bz), the class Vp,2,α,β(ϕ) reduces to Hp(A,B,α,β) introduced and studied by Owa [3].(iv)
Forϕ(z)=(1+(1-2γ)z)/(1-z), the class Vp,2,α,1(ϕ) reduces to the class Bp(α,γ)defined as
(9)Bp(α,γ)={f∈Ap:Re(zf′(z)pf(z)(f(z)zp)α)>γ,0≤γ<1,z∈E(zf′(z)pf(z)(f(z)zp)α)}.(v)
Forϕ(z)=(1+(1-2γ)z)/(1-z), the class Vp,2,α,0(ϕ) is defined as
(10){f∈Ap:Re(f(z)zp)α>γ,0≤γ<1,z∈E}.(vi)
V1,2,0,1(ϕ)=S*(ϕ) is investigated by Ma and Minda [2].(vii)
Forϕ(z)=(1+(1-2γ)z)/(1-z), the class V1,2,1,0(ϕ) reduces to the class
(11)Bγ={f∈A1:Ref(z)z>γ}.studied by Chen [5].We need the following results to obtain our main results.Lemma 3 (see [1]).
LetΩ be the class of analytic functions w, normalized by w(0)=0,satisfying condition |w(z)|<1.If w∈Ωand w(z)=w1z+w2z2+⋯, z∈E,then
(12)|w2-tw12|≤{-t,t<-1,1,-1≤t≤1,t,t>1.
For t≤-1 or t≥1,the equality holds, if and only if w(z)=z or one of its rotation. For -1≤t≤1,the equality holds, if w(z)=z2 or one of its rotation. the equality holds for t=-1, if and only if w(z)=z((λ+z)/(1+λz))(0≤λ≤1) or one of its rotation, while for t=1, the equality holds, if and only if w(z)=-z((λ+z)/(1+λz))(0≤λ≤1) or one of its rotation. The above upper bound for -1<t<1 is sharp, and it can be improved as follows:
(13)|w2-tw12|+(1+t)|w1|2≤1,-1<t≤0,|w2-tw12|+(1-t)|w1|2≤1,0<t<1.Lemma 4 (see [6, (7), page 10]).
Ifw∈Ωand w(z)=w1z+w2z2+⋯, z∈E,then
(14)|w2-tw12|≤max{1;|t|},
for any complex number t. The result is sharp for the functions w(z)=z2 or w(z)=z.Lemma 5 (see [7]).
Ifw∈Ω, then for any real number q1 and q2, the following sharp estimate holds:
(15)|w3+q1w1w2+q2w13|≤H(q1,q2),
where
(16)H(q1,q2)={1,(q1,q2)∈D1∪D2,|q2|,(q1,q2)∈⋃k=37Dk,23(|q1|+1)(|q1|+13(|q1|+1+q2))1/2,(q1,q2)∈D8∪D9,13q2(q12-4q12-4q2)(q12-43(q2-1))1/2,(q1,q2)∈D10∪D11∖{±2,1},23(|q1|-1)(|q1|-13(|q1|-1-q2))1/2,(q1,q2)∈D12.
The extremal function up to the rotations is of the form
(17)w(z)=z3,w(z)=z,w(z)=w0(z)=z([(1-λ)ε2+λε1]-ε1ε2z)1-[(1-λ)ε1+λε2]z,w(z)=w1(z)=z(t1-z)1-t1z,w(z)=w2(z)=z(t2+z)1+t2z,|ε1|=|ε2|=1,ε1=t0-e-iθ0/2(a∓b),ε2=-e-iθ0/2(ia±b),a=t0cosθ02,b=1-t02sin2θ02,λ=b±a2b,t0=(2q2(q12+2)-3q123(q2-1)(q12-4q2))1/2,t1=(|q1|+13(|q1|+1+q2))1/2,t2=(|q1|-13(|q1|-1-q2))1/2,cosθ02=q12(q2(q12+8)-2(q12+2)2q2(q12+2)-3q12).
The sets Dk, k=1,2,…,12 are defined as follows:
(18)D1={(q1,q2):|q1|≤12,|q2|≤1},D2={(q1,q2):12≤|q1|≤2,427(|q1|+1)3-(|q1|+1)≤q2≤1},D3={(q1,q2):|q1|≤12,|q2|≤-1},D4={(q1,q2):|q1|≥12,q2≤-23(|q1|+1)},D5={(q1,q2):|q1|≤2,q2≥1},D6={(q1,q2):2≤|q1|≤4,q2≥112(q12+8)},D7={(q1,q2):|q1|≥4,q2≥23(|q1|-1)},D8={(q1,q2):12≤|q1|≤2,D8-23(|q1|+1)≤q2≤427(|q1|+1)3-(|q1|+1)},D9={2|q1|(|q1|+1)q12+2|q1|+4(q1,q2):|q1|≥2,-23(|q1|+1)≤q2≤2|q1|(|q1|+1)q12+2|q1|+4},D10={2|q1|(|q1|+1)q12+2|q1|+4(q1,q2):2≤|q1|≤4,2|q1|(|q1|+1)q12+2|q1|+4≤q2≤112(q12+8)},D11={2|q1|(|q1|+1)q12+2|q1|+4(q1,q2):|q1|≥4,2|q1|(|q1|+1)q12+2|q1|+4≤q2≤2|q1|(|q1|-1)q12-2|q1|+4},D12={2|q1|(|q1|-1)q12-2|q1|+4(q1,q2):|q1|≥4,2|q1|(|q1|-1)q12-2|q1|+4≤q2≤23(|q1|-1)}.


## 2. Main Results

Theorem 6.
Letϕ(z)=1+B1z+B2z2+B3z3+⋯, where Bn, s are real with B1>0 and B2≥0. Let
(19)σ1=(αp+β)2bpB12(αp+2β)×{2(B2-B1)-bpB12(α-1)(αp+2β)2(αp+β)2},σ2=(αp+β)2bpB12(αp+2β)×{2(B2+B1)-bpB12(α-1)(αp+2β)2(αp+β)2},σ3=(αp+β)2bpB12(αp+2β)×{2B2-bpB12(α-1)(αp+2β)2(αp+β)2},Φ(p,α,β,μ)=(αp+2β)(2μ+α-1)2(αp+β)2.
If f(z) is of the form (1) and belongs to the class Vp,b,α,β(ϕ), then
(20)|ap+2-μap+12|≤{bp2(αp+2β)(B2-bpB122Φ(p,α,β,μ)),μ<σ1,bpB12(αp+2β),σ1≤μ≤σ2,-bp2(αp+2β)(B2-bpB122Φ(p,α,β,μ)),μ>σ2.
Furthermore, for σ1≤μ≤σ3,
(21)|ap+2-μap+12|+1bpB1(2(1-B2B1)(αp+β)2(αp+2β)+1bpB11bpB1+bpB12(2μ+α-1)(αp+β)2(αp+2β))|ap+1|2≤bpB12(αp+2β),
and for σ3≤μ≤σ2,
(22)|ap+2-μap+12|+1bpB1(2(1+B2B1)(αp+β)2(αp+2β)bpB12(αp+2β)-bpB12(2μ+α-1)(αp+β)2(αp+2β))|ap+1|2≤bpB12(αp+2β).
For any complex number μ,
(23)|ap+2-μap+12|≤bpB12(αp+2β)max{1,|bpB12Φ(p,α,β,μ)-B2B1|}.
Also,
(24)|ap+3|≤bpB12(αp+3β)H(q1,q2),
where H(q1,q2) is defined in Lemma 3 and
(25)q1=2B2B1+bpB12(1-α)(αp+3β)(αp+β)(αp+2β),q2=B3B1+(bpB12)2(α-1)(2α-1)(αp+3β)6(αp+β)3+bpB22(1-α)(αp+3β)(αp+β)(αp+2β).
These results are sharp.Proof.
Sincef∈Vp,b,α,β(ϕ), therefore we have for a Schwarz function
(26)w(z)=w1z+w2z2+⋯,z∈E
such that
(27)1-2b+2b{(1-β)(f(z)zp)α+βzf′(z)pf(z)(f(z)zp)α}=ϕ(w(z)).
Now,
(28)1-2b+2b{(1-β)(f(z)zp)α+βzf′(z)pf(z)(f(z)zp)α}=1+2bp(αp+β)ap+1z+1bp(αp+2β)×{2ap+2+(α-1)ap+12}z2+2(αp+3β)bp×{(α-1)(α-2)6ap+3+(α-1)ap+1ap+2-2b+2b+(α-1)(α-2)6ap+13}z3+⋯.
Also, we have
(29)ϕ(w(z))=1+B1w1z+(B1w2+B1w12)z2+(B1w3+2B2w1w2+B3w13)z3+⋯.
Comparing the coefficients of z, z2, z3 and after simple calculations, we obtain
(30)ap+1=bpB1w12(αp+β),ap+2=bpB12(αp+2β)×{w2-(bpB12(α-1)(αp+2β)2(αp+β)2-B2B1)w12},ap+3=bpB12(αp+2β){w3+q1w1w2+q2w13},
where q1 and q2 are defined in (25). It can be easily followed from (30) that
(31)ap+2-μap+12=bpB12(αp+2β){w2-νw12},
where
(32)ν=bpB12Φ(p,α,β,μ)-B2B1.
The results from (20) to (22)are obtained by using Lemma 3, (23)by using Lemma 4, and (24) by using Lemma 5. To show that these results are sharp, we define the functions Kϕn(z), Fλ(z), and Gλ(z) such that
(33)Kϕn(0)=[Kϕn]′(0)-1=0,Fλ(0)=Fλ′(0)-1=0,Gλ(0)=Gλ′(0)-1=0,
with
(34)1-2b+2b{(1-β)(Kϕn(z)zp)α+βzKϕn′(z)pKϕn(z)(Kϕn(z)zp)α}≺ϕ(zn-1),1-2b+2b{(1-β)(Fλ(z)zp)α+βzFλ′(z)pFλ(z)(Fλ(z)zp)α}≺ϕ(z(z+λ)1+λz),1-2b+2b{(1-β)(Gλ(z)zp)α+βzGλ′(z)pGλ(z)(Gλ(z)zp)α}≺ϕ(-z(z+λ)1+λz).
It is clear that the functions Kϕn,Fλ,Gλ∈Vp,b,α,β(ϕ). Let Kϕ:=Kϕ2. If μ<σ1 or μ>σ2, then the equality occurs for the function Kϕ or one of its rotations. For σ1<μ<σ2, the equality is attained, if and only if f is Kϕ3 or one of its rotations. When μ=σ1, then the equality holds for the function Fλ or one of its rotations. If μ=σ2, then the equality is obtained for the function Gλ or one of its rotations.Corollary 7.
Forb=2, the results from (20) to (24) coincide with the results proved by Ramachandran et al. [8].Corollary 8.
Forb=2,  p=1, and β=1,the results from (20) to (22) coincide with the results obtained by Ravichandran et al. [4] for the class Bα(ϕ).Corollary 9.
Forb=2,  α=0, and β=1,the results from (20) to (24) coincide with the results obtained by Ali et al. [1] for the class Sp*(ϕ).Corollary 10.
Forb=2, p=1, α=0, and β=1,the results from (20) to (22) coincide with the results obtained by Ma and Minda [2] for the class S*(ϕ).


### 2.1. Application of Theorem6 to the Function Defined by Convolutions

Theorem 11.
Letϕ(z)=1+B1z+B2z2+B3z3+⋯, where Bn, s are real with B1>0 and B2≥0. Let
(35)σ1=gp+12gp+2(αp+β)2bpB12(αp+2β)×{(α-1)(αp+2β)2(αp+β)22(B2-B1)gp+12gp+2-bpB12(α-1)(αp+2β)2(αp+β)2},σ2=gp+12gp+2(αp+β)2bpB12(αp+2β)×{(α-1)(αp+2β)2(αp+β)22(B2+B1)gp+12gp+2-bpB12(α-1)(αp+2β)2(αp+β)2},σ3=gp+12gp+2(αp+β)2bpB12(αp+2β)×{2B2-bpB12(α-1)(αp+2β)2(αp+β)2},Φ*(p,α,β,μ)=(αp+2β)(2μ(gp+2/gp+1)+α-1)2(αp+β)2.
If f(z) is of the form (1) and belongs to the class Vp,b,α,β,g(ϕ), then
(36)|ap+2-μap+12|≤{bp2gp+2(αp+2β)(B2-bpB122Φ*(p,α,β,μ)),μ<σ1,bpB12gp+2(αp+2β),σ1≤μ≤σ2,bp2gp+2(αp+2β)(bpB122Φ*(p,α,β,μ)-B2),μ>σ2.
Furthermore, for σ1≤μ≤σ3,
(37)|ap+2-μap+12|+gp+12bpB1gp+2(2(1-B2B1)(αp+β)2(αp+2β)+gp+12bpB1gp+2B2B1+bpB12(2μgp+2gp+1+α-1)(αp+β)2(αp+2β))|ap+1|2≤bpB12gp+2(αp+2β),
and for σ3≤μ≤σ2,
(38)|ap+2-μap+12|+gp+12bpB1gp+2(2(1+B2B1)(αp+β)2(αp+2β)+gp+12bpB1gp+2(1+B2B1)-bpB12(2μgp+2gp+1+α-1)(αp+β)2(αp+2β))|ap+1|2≤bpB12gp+2(αp+2β).
For any complex number μ,
(39)|ap+2-μap+12|≤bpB12gp+2(αp+2β)max{1,|bpB12Φ*(p,α,β,μ)-B2B1|}.
Also,
(40)|ap+3|≤bpB12(αp+3β)gp+3H(q1,q2),
where H(q1,q2)is defined in Lemma 5 and
(41)q1=2B2B1+bpB12(1-α)(αp+3β)(αp+β)(αp+2β),q2=B3B1+(bpB12)2(α-1)(2α-1)(αp+3β)6(αp+β)3+bpB22(1-α)(αp+3β)(αp+β)(αp+2β).
These results are sharp.Proof.
Sincef∈Vp,b,α,β,g(ϕ), therefore we have for a Schwarz function w,such that
(42)1-2b+2b{βz(f*g)′(z)p(f*g)(z)(1-β)((f*g)(z)zp)α1-2b+2b+βz(f*g)′(z)p(f*g)(z)((f*g)(z)zp)α}=ϕ(w(z)).
Now,
(43)1-2b+2b{βz(f*g)′(z)p(f*g)(z)(1-β)((f*g)(z)zp)α1-2b+2b+βz(f*g)′(z)p(f*g)(z)((f*g)(z)zp)α}=1+2bp(αp+β)ap+1gp+1z+1bp(αp+2β)×{2ap+2gp+2+(α-1)ap+12gp+12}z2+2bp(αp+3β)×{ap+3gp+3+(α-1)ap+1ap+2gp+1gp+2+(α-1)(α-2)6ap+13gp+13}z3+⋯.
Also, we obtain
(44)ϕ(w(z))=1+B1w1z+(B1w2+B1w12)z2+(B1w3+2B2w1w2+B3w13)z3+⋯.
Comparing the coefficients of z, z2, z3 and after simple calculations, we obtain
(45)ap+1=bpB1w12gp+1(αp+β),ap+2=bpB12gp+2(αp+2β)×{w2-(bpB12(α-1)(αp+2β)2(αp+β)2-B2B1)w12},ap+3=bpB12gp+3(αp+2β){w3+q1w1w2+q2w13}.
The remaining proof of the theorem is similar to the proof of Theorem 6.Corollary 12.
Forb=2, the results from (36) to (40) coincide with the results proved by Ramachandran et al. [8] for the class Rp,1,α,β,g(ϕ).Corollary 13.
Forb=2, α=0, and β=1,the results from (36) to (40) coincide with the results obtained by Ali et al. [1] for the class Sp,g*(ϕ).Corollary 14.
Forb=2, p=1, α=0, and β=1,
(46)g2=Γ(3)Γ(2-λ)Γ(3-λ)=22-λ,g3=Γ(4)Γ(2-λ)Γ(4-λ)=6(2-λ)(3-λ),B1=8π2,B2=163π2,
the results from (36) to (38) coincide with the results obtained by Srivastava and Mishra [9].


## 2.1. Application of Theorem6 to the Function Defined by Convolutions

Theorem 11.
Letϕ(z)=1+B1z+B2z2+B3z3+⋯, where Bn, s are real with B1>0 and B2≥0. Let
(35)σ1=gp+12gp+2(αp+β)2bpB12(αp+2β)×{(α-1)(αp+2β)2(αp+β)22(B2-B1)gp+12gp+2-bpB12(α-1)(αp+2β)2(αp+β)2},σ2=gp+12gp+2(αp+β)2bpB12(αp+2β)×{(α-1)(αp+2β)2(αp+β)22(B2+B1)gp+12gp+2-bpB12(α-1)(αp+2β)2(αp+β)2},σ3=gp+12gp+2(αp+β)2bpB12(αp+2β)×{2B2-bpB12(α-1)(αp+2β)2(αp+β)2},Φ*(p,α,β,μ)=(αp+2β)(2μ(gp+2/gp+1)+α-1)2(αp+β)2.
If f(z) is of the form (1) and belongs to the class Vp,b,α,β,g(ϕ), then
(36)|ap+2-μap+12|≤{bp2gp+2(αp+2β)(B2-bpB122Φ*(p,α,β,μ)),μ<σ1,bpB12gp+2(αp+2β),σ1≤μ≤σ2,bp2gp+2(αp+2β)(bpB122Φ*(p,α,β,μ)-B2),μ>σ2.
Furthermore, for σ1≤μ≤σ3,
(37)|ap+2-μap+12|+gp+12bpB1gp+2(2(1-B2B1)(αp+β)2(αp+2β)+gp+12bpB1gp+2B2B1+bpB12(2μgp+2gp+1+α-1)(αp+β)2(αp+2β))|ap+1|2≤bpB12gp+2(αp+2β),
and for σ3≤μ≤σ2,
(38)|ap+2-μap+12|+gp+12bpB1gp+2(2(1+B2B1)(αp+β)2(αp+2β)+gp+12bpB1gp+2(1+B2B1)-bpB12(2μgp+2gp+1+α-1)(αp+β)2(αp+2β))|ap+1|2≤bpB12gp+2(αp+2β).
For any complex number μ,
(39)|ap+2-μap+12|≤bpB12gp+2(αp+2β)max{1,|bpB12Φ*(p,α,β,μ)-B2B1|}.
Also,
(40)|ap+3|≤bpB12(αp+3β)gp+3H(q1,q2),
where H(q1,q2)is defined in Lemma 5 and
(41)q1=2B2B1+bpB12(1-α)(αp+3β)(αp+β)(αp+2β),q2=B3B1+(bpB12)2(α-1)(2α-1)(αp+3β)6(αp+β)3+bpB22(1-α)(αp+3β)(αp+β)(αp+2β).
These results are sharp.Proof.
Sincef∈Vp,b,α,β,g(ϕ), therefore we have for a Schwarz function w,such that
(42)1-2b+2b{βz(f*g)′(z)p(f*g)(z)(1-β)((f*g)(z)zp)α1-2b+2b+βz(f*g)′(z)p(f*g)(z)((f*g)(z)zp)α}=ϕ(w(z)).
Now,
(43)1-2b+2b{βz(f*g)′(z)p(f*g)(z)(1-β)((f*g)(z)zp)α1-2b+2b+βz(f*g)′(z)p(f*g)(z)((f*g)(z)zp)α}=1+2bp(αp+β)ap+1gp+1z+1bp(αp+2β)×{2ap+2gp+2+(α-1)ap+12gp+12}z2+2bp(αp+3β)×{ap+3gp+3+(α-1)ap+1ap+2gp+1gp+2+(α-1)(α-2)6ap+13gp+13}z3+⋯.
Also, we obtain
(44)ϕ(w(z))=1+B1w1z+(B1w2+B1w12)z2+(B1w3+2B2w1w2+B3w13)z3+⋯.
Comparing the coefficients of z, z2, z3 and after simple calculations, we obtain
(45)ap+1=bpB1w12gp+1(αp+β),ap+2=bpB12gp+2(αp+2β)×{w2-(bpB12(α-1)(αp+2β)2(αp+β)2-B2B1)w12},ap+3=bpB12gp+3(αp+2β){w3+q1w1w2+q2w13}.
The remaining proof of the theorem is similar to the proof of Theorem 6.Corollary 12.
Forb=2, the results from (36) to (40) coincide with the results proved by Ramachandran et al. [8] for the class Rp,1,α,β,g(ϕ).Corollary 13.
Forb=2, α=0, and β=1,the results from (36) to (40) coincide with the results obtained by Ali et al. [1] for the class Sp,g*(ϕ).Corollary 14.
Forb=2, p=1, α=0, and β=1,
(46)g2=Γ(3)Γ(2-λ)Γ(3-λ)=22-λ,g3=Γ(4)Γ(2-λ)Γ(4-λ)=6(2-λ)(3-λ),B1=8π2,B2=163π2,
the results from (36) to (38) coincide with the results obtained by Srivastava and Mishra [9].



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*Source: 127615-2013-06-12.xml*