# Some Properties of Subclasses of Multivalent Functions

Authors: Muhammet  Kamali; Fatma Sağsöz
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The authors introduce two new subclasses denoted byℑ∗(Ω,λ,p,α) and ℑε∗(Ω,λ,p,α) of the class A(p,n) of p-valent analytic functions. They obtain coefficient inequality for the class ℑ∗(Ω,λ,p,α). They investigate various properties of classes ℑ∗(Ω,λ,p,α) and ℑε∗(Ω,λ,p,α). Furthermore, they derive partial sums associated with the class ℑε∗(Ω,λ,p,α).


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## Body

## 1. Introduction and  Definition

LetA(p,n) denote the class of functions of the form (1.1)f(z)=zp+∑k=n∞ap+kzp+k(n,p∈N={1,2,…}),
which are analytic and p-valent  in the open unit disc U={z:z∈ℂ,|z|<1}. We write A(1,1)=A.A functionf∈A(p,n) is said to be in the class  S(p,n,α) of  p-valently  star-like functions  of  order α if  it satisfies the condition(1.2)Re(zf′(z)f(z))>α(z∈C;0≤α<p).
Furthermore, a function f∈A(p,n) is said to be in the class  K(p,n,α) of  p-valently convex functions  of  order α if  it satisfies the condition(1.3)Re(1+zf′′(z)f′(z))>α(z∈U;0≤α<p).
The classes S(p,n,α) and K(p,n,α)  were studied by Owa [1]. The class S*(p,α)∶=S(p,1,α) was considered by Patil and Thakare [2].We denote byT(p,n) the subclass of the class A(p,n) consisting of functions of the form (1.4)f(z)=zp-∑k=n∞ap+kzp+k(ap+k≥0;n,p∈N)
and define two further classes T*(p,n,α) and  C(p,n,α) by(1.5)T*(p,n,α)∶=S(p,n,α)∩T(p,n),C(p,n,α)∶=K(p,n,α)∩T(p,n).
For the classes T*(p,α)∶=S*(p,α)∩T(p),C(p,α)∶=K(p,α)∩T(p).The following lemmas were given by Owa [3].Lemma 1.1.
Let the functionf be defined by
(1.6)f(z)=zp-∑k=1∞ap+kzp+k(ap+k≥0;p∈N).
Then, f is in the class T*(p,α) if and only if
(1.7)∑k=1∞(p+k-α)ap+k≤p-α.
The result is sharp.Lemma 1.2.
Let the functionf be defined  by (1.6).  Then, f is in the class C(p,α) if and only if
(1.8)∑k=1∞(p+k)(p+k-α)ap+k≤p(p-α).
The result is sharp.For a functionf  defined by  (1.6) and in the class T*(p,α), Lemma 1.1 yields(1.9)ap+1≤p-αp+1-α.
On the other hand, for a function f defined by (1.6) and in the class C(p,α), Lemma 1.2 yields(1.10)ap+1≤p(p-α)(p+1)(p+1-α).
In view of the coefficient inequalities (1.9) and (1.10), it would seem to be natural to introduce and study here two further classes Tε*(p,α) and  Cε(p,α) of analytic and p-valent functions, where Tε*(p,α) denotes the subclass of T*(p,α) consisting of functions of the form(1.11)f(z)=zp-(p-α)εp+1-αzp+1-∑k=2∞ap+kzp+k(ap+k≥0,p∈N,k∈N-{1};0≤α<p;0≤ε<1)
and  Cε(p,α) denotes the subclass of  C(p,α) consisting of functions of the form(1.12)f(z)=zp-p(p-α)ε(p+1)(p+1-α)zp+1-∑k=2∞ap+kzp+k(ap+k≥0,p∈N,k∈N-{1};0≤α<p;0≤ε<1).
The classes  Tε*(p,α) and  Cε(p,α) are studied by Aouf et al. [4].The classes(1.13)Tε*(α)∶=Tε*(1,α),Cε(α)∶=Cε(1,α)were considered earlier by Silverman and Silvia [5].Now, we  give the following equalities for the functionsf(z) belonging to the class A(p,n):(1.14)D0f(z)=f(z),D1f(z)=Df(z)=z(D0f(z))′=z[pzp-1+∑k=n∞(p+k)ap+kzp+k-1]=pzp+∑k=n∞(p+k)ap+kzp+k,D2f(z)=D(Df(z))=z(D1f(z))′=z[pzp+∑k=n∞(p+k)ap+kzp+k]′=p2zp+∑k=n∞(p+k)2ap+kzp+k,⋮DΩf(z)=D(DΩ-1f(z))=pΩzp+∑k=n∞(k+p)Ωap+kzp+k.
We define  ℘:A(p,n)→A(p,n) such that(1.15)℘(Ω,λ,p)=(1pΩ-λ)DΩf(z)+λpz(DΩf(z))′(0≤λ≤1pΩ,Ω∈N∪{0}).A functionf(z)∈A(p,n) is said to be in the class ℑ(Ω,λ,p,α) if it satisfies the inequality(1.16)Re{z(℘(Ω,λ,p))′℘(Ω,λ,p)}=Re{z(1/pΩ+(1/p-1)λ)(DΩf(z))′+(λ/p)z(DΩf(z))′′(1/pΩ-λ)DΩf(z)+(λ/p)z(DΩf(z))′}>α,
for some  α(0≤α<p),0≤λ≤1/pΩ,  Ω∈ℕ∪{0} and  for all  z∈U.IfΩ=0 and λ=0, we obtain the condition (1.2). Furthermore, we obtain the condition (1.3) for Ω=0 and λ=1.We denote byT(p,n) the subclass of the class A(p,n) consisting of functions of the form (1.17)f(z)=zp-∑k=n∞ak+pzk+p(ak+p≥0;n,p∈N),
and  define the class ℑ*(Ω,λ,p,α) by(1.18)I*(Ω,λ,p,α)=I(Ω,λ,p,α)∩T(p,n).Furthermore, we denote byℑε*(Ω,λ,p,α) the subclass of ℑ*(Ω,λ,p,α) consisting of functions of the form
(1.19)f(z)=zp-(p-α)ε((p+n)/p)Ω(1+λnpΩ-1)(n+p-α)zp+n-∑k=n+1∞ap+kzp+k(0≤ε<1).
The main object of the present paper is to investigate interesting properties and characteristics of the classes ℑ*(Ω,λ,p,α) and ℑε*(Ω,λ,p,α). Also, the partial sums is defined for f function defined by (1.19).


## 2. A Coefficient Inequality   for  the Classℑ*(Ω,λ,p,α) and   Some Theorems  for the Class ℑε*(Ω,λ,p,α)

First,  we give a coefficient inequality for the classℑ*(Ω,λ,p,α).Theorem 2.1.
Let the functionf be defined by (1.17).  Then, f is in the class  ℑ*(Ω,λ,p,α) if and only if
(2.1)∑k=n∞(k+pp)Ω(1+λkpΩ-1)(k+p-α)ap+k≤p-α.Proof.
Suppose thatf(z)∈ℑ*(Ω,λ,p,α). Then, we find from (1.16) that
(2.2)Re{z(℘(Ω,λ,p))′℘(Ω,λ,p)}=Re{z(1/pΩ+(1/p-1)λ)(DΩf(z))′+(λ/p)z(DΩf(z))′′(1/pΩ-λ)DΩf(z)+(λ/p)z(DΩf(z))′}=Re{pzp-∑k=n∞(k+p)Ω+1(1/pΩ+λk/p)ap+kzp+kzp-∑k=n∞(k+p)Ω(1/pΩ+λk/p)ap+kzp+k}>α.
If we choose  z to be real and let z→1-, we get
(2.3){p-∑k=n∞(k+p)Ω+1(1/pΩ+λk/p)ap+k1-∑k=n∞(k+p)Ω(1/pΩ+λk/p)ap+k}≥α
or, equivalently,
(2.4)p-∑k=n∞(k+p)Ω+1(1/pΩ+λk/p)ap+k≥α{1-∑k=n∞(k+p)Ω(1/pΩ+λk/p)ap+k}.
Thus, we have
(2.5)∑k=n∞{(k+p)Ω+1(1/pΩ+λk/p)-(k+p)Ω(1/pΩ+λk/p)}ap+k≤p-α
or
(2.6)∑k=n∞{(k+pp)Ω(1+λkpΩ-1)(k+p-α)}ap+k≤p-α.
Conversely, suppose that the inequality (2.1) holds true and let
(2.7)z∈∂U={z:z∈C,|z|=1}.
Then, we find from the definition  (1.4) that
(2.8)|z(℘(Ω,λ,p))′℘(Ω,λ,p)-p|=|z(1/pΩ+(1/p-1)λ)(DΩf(z))′+(λ/p)z(DΩf(z))′′(1/pΩ-λ)DΩf(z)+(λ/p)z(DΩf(z))′-p|=|pzp-∑k=n∞((k+p)/p)Ω(k+p)(1+λkpΩ-1)ap+kzp+kzp-∑k=n∞((k+p)/p)Ω(1+λkpΩ-1)ap+kzp+k-p|≤∑k=n∞k((k+p)/p)Ω(1+λkpΩ-1)ap+k1-∑k=n∞((k+p)/p)Ω(1+λkpΩ-1)ap+k.
By means of inequality (2.1), we can write
(2.9)∑k=n∞(k+pp)Ω(1+λkpΩ-1)(k+p-α)ap+k≤p-α⟹∑k=n∞k(k+pp)Ω(1+λkpΩ-1)ap+k≤p-α-∑k=n∞(k+pp)Ω(1+λkpΩ-1)(p-α)ap+k
or
(2.10)∑k=n∞k(k+pp)Ω(1+λkpΩ-1)ap+k≤p-α-(p-α)∑k=n∞(k+pp)Ω(1+λkpΩ-1)ap+k.
Thus, we obtain
(2.11)|z(1/pΩ+(1/p-1)λ)(DΩf(z))′+(λ/p)z(DΩf(z))′′(1/pΩ-λ)DΩf(z)+(λ/p)z(DΩf(z))′-p|=|∑k=n∞k((k+p)/p)Ω(1+λkpΩ-1)ap+kzk1-∑k=n∞((k+p)/p)Ω(1+λkpΩ-1)ap+kzk|≤p-α-(p-α)∑k=n∞((k+p)/p)Ω(1+λkpΩ-1)ap+k1-∑k=n∞((k+p)/p)Ω(1+λkpΩ-1)ap+k=p-α.
This evidently completes the proof of Theorem 2.1.Now, we give a characterization theorem for the classℑε*(Ω,λ,p,α).Theorem 2.2.
Let the functionf be defined by (1.19). Then, f is in the class ℑε*(Ω,λ,p,α) if and only if
(2.12)∑k=n+1∞(k+pp)Ω(1+λkpΩ-1)(k+p-α)ap+k≤(p-α)(1-ε)(0≤α<p,0≤ε<1,0≤λ≤1pΩ).
The result is sharp for the function f given by
(2.13)f(z)=zp-(p-α)ε((p+n)/p)Ω(1+λnpΩ-1)(n+p-α)zp+n-(p-α)(1-ε)((p+k)/p)Ω(1+λkpΩ-1)(k+p-α)zp+k(k=n+1,n+2,…;n∈N).Proof.
Using inequality (2.1), we have
(2.14)(n+pp)Ω(1+λnpΩ-1)(n+p-α)ap+n+∑k=n+1∞(k+pp)Ω(1+λkpΩ-1)(k+p-α)ap+k≤p-α
or
(2.15)∑k=n+1∞(k+pp)Ω(1+λkpΩ-1)(k+p-α)ap+k≤p-α-(n+pp)Ω(1+λnpΩ-1)(n+p-α)ap+n.
Thus, by setting
(2.16)ap+n=(p-α)ε((p+n)/p)Ω(1+λnpΩ-1)(n+p-α),
we obtain
(2.17)∑k=n+1∞(k+pp)Ω(1+λkpΩ-1)(k+p-α)ap+k≤p-α-(n+pp)Ω(1+λnpΩ-1)(n+p-α)(p-α)ε((p+n)/p)Ω(1+λnpΩ-1)(n+p-α)
or
(2.18)∑k=n+1∞(k+pp)Ω(1+λkpΩ-1)(k+p-α)ap+k≤(p-α)(1-ε).
A closure theorem for the class  ℑε*(Ω,λ,p,α) is given by the following.Theorem 2.3.
Let(2.19)fs(z)=zp-(p-α)ε((p+n)/p)Ω(1+λnpΩ-1)(n+p-α)zp+n-∑k=n+1∞ap+k,szp+k(ap+k,s≥0;p,n∈N;0≤ε<1;s=1,2,3,…,m).
If  fs∈ℑε*(Ω,λ,p,α), then the function g given by
(2.20)g(z)=zp-(p-α)ε((p+n)/p)Ω(1+λnpΩ-1)(n+p-α)zp+n-∑k=n+1∞bp+kzp+k,
with
(2.21)bp+k∶=1m∑s=1map+k,s≥0,
is also in the class ℑε*(Ω,λ,p,α).Proof.
Sincefs∈ℑε*(Ω,λ,p,α) for  s=1,2,…,m, it follows from Theorem 2.2  that
(2.22)∑k=n+1∞(k+pp)Ω(1+λkpΩ-1)(k+p-α)ap+k,s≤(p-α)(1-ε)(s=1,2,…,m).
By applying (2.22) and the definition (2.21), we write
(2.23)∑k=n+1∞(k+pp)Ω(1+λkpΩ-1)(k+p-α)bp+k=∑k=n+1∞(k+pp)Ω(1+λkpΩ-1)(k+p-α){1m∑s=1map+k,s}=1m∑s=1m{∑k=n+1∞(k+pp)Ω(1+λkpΩ-1)(k+p-α)ap+k,s}≤1m∑s=1m(p-α)(1-ε)=(p-α)(1-ε).Theorem 2.4.
Let(2.24)fp+n(z)=zp-(p-α)ε((p+n)/p)Ω(1+λnpΩ-1)(n+p-α)zp+n,fp+k(z)=zp-(p-α)ε((p+n)/p)Ω(1+λnpΩ-1)(n+p-α)zp+n-(p-α)(1-ε)((p+k)/p)Ω(1+λkpΩ-1)(k+p-α)zp+k,
where k∈{n+1,n+2,…}.
Then,f is in the class  ℑε*(Ω,λ,p,α) if and only if it can be expressed in the form
(2.25)f(z)=∑k=n∞ηp+kfp+k(z)(ηp+k≥0;∑k=n∞ηp+k=1).Proof.
Suppose thatf is given by (2.25), so that we find from (2.24) that
(2.26)f(z)=zp-(p-α)ε((p+n)/p)Ω(1+λnpΩ-1)(n+p-α)zp+n-∑k=n+1∞(p-α)(1-ε)((p+k)/p)Ω(1+λkpΩ-1)(k+p-α)ηp+kzp+k,
where the coefficients  ηp+k are given with  ∑k=n∞ηp+k=1,  ηp+k≥0. Then, since,
(2.27)∑k=n+1∞(k+pp)Ω(1+λkpΩ-1)(k+p-α)(p-α)(1-ε)((k+p)/p)Ω(1+λkpΩ-1)(k+p-α)ηp+k=(p-α)(1-ε)∑k=n+1∞ηp+k=(p-α)(1-ε)(1-ηp+n)≤(p-α)(1-ε),
we conclude from Theorem 2.2 that  f∈ℑε*(Ω,λ,p,α).
Conversely, let us assume that the functionf defined by (1.19) is in the class  ℑε*(Ω,λ,p,α). Then,
(2.28)ap+k≤(p-α)(1-ε)((k+p)/p)Ω(1+λkpΩ-1)(k+p-α),
which follows readily from (2.12) for k∈{n+1,n+2,…}.
Fork∈{n+1,n+2,…}, setting
(2.29)ηp+k=((k+p)/p)Ω(1+λkpΩ-1)(k+p-α)(p-α)(1-ε)ap+k
and ηp+n=1-∑k=n+1∞ηp+k, we thus arrive at (2.25). This  completes the proof of Theorem 2.4.Theorem 2.5.
Letf be given by  (1.19) and define the partial sums sm(z) by
(2.30)sm(z)={zp-pΩ(p-α)ε(p+n)Ω(1+λnpΩ-1)(n+p-α)zp+n,m=n,zp-pΩ(p-α)ε(p+n)Ω(1+λnpΩ-1)(n+p-α)zp+n-∑k=n+1map+kzp+k,m=n+1,n+2,….
Suppose also that
(2.31)∑k=n+1∞dp+kap+k≤1-pΩ(p-α)ε(p+n)Ω(1+λnpΩ-1)(n+p-α),(wheredp+k=(p+k)Ω(1+λkpΩ-1)(k+p-α)pΩ(p-α)(1-ε);0≤α<p,0≤ε<1,0≤λ≤1pΩ).
Then,   for m≥n+1, one has
(2.32)Re{f(z)sm(z)}>1-pΩ(p-α)(1-ε)(p+m+1)Ω[1+λ(m+1)pΩ-1](p+m+1-α),(2.33)Re(sm(z)f(z))>(p+m+1)Ω{1+λ(m+1)pΩ-1}(m+p+1-α)pΩ(p-α)(1-ε)+(p+m+1)Ω{1+λ(m+1)pΩ-1}(m+p+1-α).
Each of the bounds in (2.32) and (2.33) is the best  possible.Proof.
From (2.31) and Theorem 2.2, we have that  f∈ℑε*(Ω,λ,p,α). By definition dp+k, we can write
(2.34)dp+k=((p+k)/p)Ω(1+λkpΩ-1)(k+p-α)(p-α)(1-ε)=(1+kp)Ω(1+λkpΩ-1)[p-α+k(p-α)(1-ε)]=(1+kp)Ω(1+λkpΩ-1)[11-ε+k(p-α)(1-ε)]>1.
Under the hypothesis of this theorem, we can see from (2.31) that  dp+k+1>dp+k>1, for    k=n+1,n+2…. Therefore, we have
(2.35)∑k=n+1map+k+dp+m+1∑k=m+1∞ap+k≤∑k=n+1∞dp+kap+k≤1-pΩ(p-α)ε(p+n)Ω(1+λnpΩ-1)(p+n-α).
Using (1.19) and (2.30), we can write
(2.36)f(z)sm(z)=zp-(pΩ(p-α)ε/(p+n)Ω(1+λnpΩ-1)(n+p-α))zp+n-∑k=n+1∞ap+kzp+kzp-(pΩ(p-α)ε/(p+n)Ω(1+λnpΩ-1)(n+p-α))zp+n-∑k=n+1map+kzp+k=1-∑k=m+1∞ap+kzp+kzp-(pΩ(p-α)ε/(p+n)Ω(1+λnpΩ-1)(n+p-α))zp+n-∑k=n+1map+kzp+k.
Set
(2.37)Ψ1(z)=dp+m+1[f(z)sm(z)-(1-1dp+m+1)]=1-dp+m+1∑k=m+1∞ap+kzk1-(pΩ(p-α)ε/(p+n)Ω(1+λnpΩ-1)(n+p-α))zn-∑k=n+1map+kzk.
By applying (2.35) and (2.37), we find that
(2.38)|ψ1(z)-1ψ1(z)+1|=|-dp+m+1∑k=m+1∞ap+kzk2-(2pΩ(p-α)ε(p+n)Ω(1+λnpΩ-1)(n+p-α))zn-2∑k=n+1map+kzk-dp+m+1∑k=m+1∞ap+kzk|≤dp+m+1∑k=m+1∞ap+kzp+k2[1-(pΩ(p-α)ε(p+n)Ω(1+λnpΩ-1)(n+p-α))-∑k=n+1map+k]-dp+m+1∑k=m+1∞ap+k≤1(z∈U;m≥n+1),
which shows that  ReΨ1(z)>0. Thus, we obtain
(2.39)ReΨ1(z)=Re{dp+m+1[f(z)sm(z)-(1-1dp+m+1)]}>0⟹Re{f(z)sm(z)}>1-1dp+m+1
or
(2.40)Re{f(z)sm(z)}>1-pΩ(p-α)(1-ε)(p+m+1)Ω[1+λ(m+1)pΩ-1](p+m+1-α).
Let
(2.41)f(z)=zp-pΩ(p-α)ε(p+n)Ω(1+λnpΩ-1)(n+p-α)zp+n-1-pΩ(p-α)ε/(p+n)Ω(1+λnpΩ-1)(n+p-α)dp+m+1zp+m+1.
Then, f(z) satisfies the condition (2.31) and f∈ℑε*(Ω,λ,p,α). Thus, we can write
(2.42)f(z)sm(z)=1-(pΩ(p-α)ε/(p+n)Ω(1+λnpΩ-1)(n+p-α))zn1-(pΩ(p-α)ε/(p+n)Ω(1+λnpΩ-1)(n+p-α))zn-(1-(pΩ(p-α)ε/(p+n)Ω(1+λnpΩ-1)(n+p-α))/dp+m+1)zm+11-(pΩ(p-α)ε/(p+n)Ω(1+λnpΩ-1)(n+p-α))zn=1-1-((pΩ(p-α)ε/(p+n)Ω(1+λnpΩ-1)(n+p-α))/dp+m+1)zm+11-(pΩ(p-α)ε/(p+n)Ω(1+λnpΩ-1)(n+p-α))zn
and taking as z→1-(2.43)⟶1-1dp+m+1=1-pΩ(p-α)(1-ε)(p+m+1)Ω[1+λ(m+1)pΩ-1](p+m+1-α),
which shows that the bound in (2.32) is the best possible.
By using  definitions off(z) and sm(z), we can write
(2.44)sm(z)f(z)=zp-((p-α)ε/((p+n)/p)Ω(1+λnpΩ-1)(n+p-α))zp+n-∑k=n+1map+kzp+kzp-((p-α)ε/((p+n)/p)Ω(1+λnpΩ-1)(n+p-α))zp+n-∑k=n+1∞ap+kzp+k=zp-((p-α)ε/((p+n)/p)Ω(1+λnpΩ-1)(n+p-α))zp+nzp-((p-α)ε/((p+n)/p)Ω(1+λnpΩ-1)(n+p-α))zp+n-∑k=n+1∞ap+kzp+k-∑k=n+1∞ap+kzp+k+∑k=m+1∞ap+kzp+kzp-((p-α)ε/((p+n)/p)Ω(1+λnpΩ-1)(n+p-α))zp+n-∑k=n+1∞ap+kzp+k=1+∑k=m+1∞ap+kzk1-((p-α)ε/((p+n)/p)Ω(1+λnpΩ-1)(n+p-α))zn-∑k=n+1∞ap+kzk.
If  we put
(2.45)ψ2(z)=(1+dp+m+1){sm(z)f(z)-dp+m+11+dp+m+1}=1+((1+dp+m+1)∑k=m+1∞ap+kzk1-((p-α)ε/((p+n)/p)Ω(1+λnpΩ-1)(n+p-α))zn-∑k=n+1∞ap+kzk)
and  make use of (2.35), we can deduce that
(2.46)|ψ2(z)-1ψ2(z)+1|=|(1+dp+m+1)∑k=m+1∞ap+kzk2-(2(p-α)ε(p+np)Ω(1+λnpΩ-1)(n+p-α))zn-2∑k=n+1map+kzk+(dp+m+1-1)∑k=m+1∞ap+kzk|≤(1+dp+m+1)∑k=m+1∞ap+k2-(2(p-α)ε(p+np)Ω(1+λnpΩ-1)(n+p-α))-2∑k=n+1map+k-(dp+m+1-1)∑k=m+1∞ap+k=(1+dp+m+1)∑k=m+1∞ap+k2{1-((p-α)ε(p+np)Ω(1+λnpΩ-1)(n+p-α))-∑k=n+1map+k}-(dp+m+1-1)∑k=m+1∞ap+k≤(1+dp+m+1)∑k=m+1∞ap+k2dp+m+1∑k=m+1∞ap+k-(dp+m+1-1)∑k=m+1∞ap+k=1,|(ψ2(z)-1)/(ψ2(z)+1)|≤1 requires that Re{ψ2(z)}>0. Thus, we obtain
(2.47)Re{Ψ2(z)}=(1+dp+m+1)Re[sm(z)f(z)-(dp+m+11+dp+m+1)]>0
or
(2.48)Re{sm(z)f(z)}>(dp+m+11+dp+m+1).
It follows from the last inequality that assertion (2.33) of the Theorem 2.5 holds.
The bound in  (2.33) is sharp with the extremal function given by  (2.41). The proof of the theorem is thus completed.IfΩ=0, λ=0, and  n=1 are taken in Theorem 2.5, the following result is obtained  given  by  Liu [6].Corollary 2.6.
Letf be given by  (1.19) and define the partial sums sm(z) by
(2.49)sm(z)={zp-(p-α)εp+1-αzp+1,m=1,zp-(p-α)ε(n+p-α)zp+1-∑k=2map+kzp+k,m=2,3,….
Suppose also that
(2.50)∑k=2∞dp+kap+k≤1-(p-α)εp+1-α(wheredp+k=k+p-α(p-α)(1-ε);0≤α<p,0≤ε<1).
Then,   for m≥2, one has
(2.51)Re{f(z)sm(z)}>(p-α)ε+m+1p+m+1-αRe(sm(z)f(z))>p+m+1-α(p-α)(2-ε)+m+1.
Each of the bounds in  (2.51) is the best  possible.



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*Source: 361647-2011-02-08.xml*