# The Fekete-Szegö Problem forp-Valently Janowski Starlike and Convex Functions

Authors: Toshio Hayami; Shigeyoshi Owa
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

Forp-valently Janowski starlike and convex functions defined by applying subordination for the generalized Janowski function, the sharp upper bounds of a functional |ap+2−μa2p+1| related to the Fekete-Szegö problem are given.


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## Body

## 1. Introduction

Let𝒜p denote the family of functions f(z) normalized by
(1.1)f(z)=zp+∑n=p+1∞anzn(p=1,2,3,…),
which are analytic in the open unit disk 𝕌={z∈ℂ:|z|<1}. Furtheremore, let 𝒲 be the class of functions w(z) of the form
(1.2)w(z)=∑k=1∞wkzk,
which are analytic and satisfy |w(z)|<1 in 𝕌. Then, a function w(z)∈𝒲 is called the Schwarz function. If f(z)∈𝒜p satisfies the following condition
(1.3)Re[1+1b(zf′(z)f(z)-p)]>0(z∈U),
for some complex number b(b≠0), then f(z) is said to be p-valently starlike function of complex order b. We denote by 𝒮b*(p) the subclass of 𝒜p consisting of all functions f(z) which are p-valently starlike functions of complex order b. Similarly, we say that f(z) is a member of the class 𝒦b(p) of p-valently convex functions of complex order b in 𝕌 if f(z)∈𝒜p satisfies
(1.4)Re[1+1b(zf′′(z)f′(z)-(p-1))]>0(z∈U),
for some complex number b(b≠0).Next, letF(z)=zf′(z)/f(z)=u+iv and b=ρeiφ(ρ>0,0≦φ<2π). Then, the condition of the definition of 𝒮b*(p) is equivalent to
(1.5)Re[1+1b(zf′(z)f(z)-p)]=1+cos⁡φρ(u-p)+sinφρv>0.We denote byd(l1,p) the distance between the boundary line l1:(cosφ)u+(sinφ)v+ρ-pcosφ=0 of the half plane satisfying the condition (1.5) and the point F(0)=p. A simple computation gives us that
(1.6)d(l1,p)=|cos⁡φ×p+sinφ×0+ρ-pcos⁡φ|cos⁡2φ+sin2φ=ρ,
that is, that d(l1,p) is always equal to |b|=ρ regardless of φ. Thus, if we consider the circle C1 with center at p and radius ρ, then we can know the definition of 𝒮b*(p) means that F(𝕌) is covered by the half plane separated by a tangent line of C1 and containing C1. For p=1, the same things are discussed by Hayami and Owa [3].Then, we introduce the following function:(1.7)p(z)=1+Az1+Bz(-1≦B<A≦1),
which has been investigated by Janowski [4]. Therefore, the function p(z) given by (1.7) is said to be the Janowski function. Furthermore, as a generalization of the Janowski function, Kuroki et al. [6] have investigated the Janowski function for some complex parameters A and B which satisfy one of the following conditions:
(1.8)(i)A≠B,|B|<1,|A|≦1,Re(1-AB¯)≧|A-B|;(ii)A≠B,|B|=1,|A|≦1,1-AB¯>0.
Here, we note that the Janowski function generalized by the conditions (1.8) is analytic and univalent in 𝕌, and satisfies Re(p(z))>0(z∈𝕌). Moreover, Kuroki and Owa [5] discussed the fact that the condition |A|≦1 can be omitted from among the conditions in (1.8)-(i) as the conditions for A and B to satisfy Re(p(z))>0. In the present paper, we consider the more general Janowski function p(z) as follows:
(1.9)p(z)=p+Az1+Bz(p=1,2,3,…),
for some complex parameter A and some real parameter B(A≠pB,-1≦B≦0). Then, we don't need to discuss the other cases because for the function:
(1.10)q(z)=p+A1z1+B1z(A1,B1∈C,A1≠pB1,|B1|≦1),
letting B1=|B1|eiθ and replacing z by -e-iθz in (1.10), we see that
(1.11)p(z)=q(-e-iθz)=p-A1e-iθz1-|B1|z≡p+Az1+Bz(A=-A1e-iθ,B=-|B1|),
maps 𝕌 onto the same circular domain as q(𝕌).Remark 1.1.
For the caseB=-1 in (1.9), we know that p(z) maps 𝕌 onto the following half plane:
(1.12)Re(p+A¯)p(z)>p2-|A|22,
and for the case -1<B≦0 in (1.9), p(z) maps 𝕌 onto the circular domain
(1.13)|p(z)-p+AB1-B2|<|A+pB|1-B2.Letp(z) and q(z) be analytic in 𝕌. Then, we say that the function p(z) is subordinate to q(z) in 𝕌, written by
(1.14)p(z)≺q(z)(z∈U),
if there exists a function w(z)∈𝒲 such that p(z)=q(w(z))(z∈𝕌). In particular, if q(z) is univalent in 𝕌, then p(z)≺q(z) if and only if
(1.15)p(0)=q(0),p(U)⊂q(U).We next define the subclasses of𝒜p by applying the subordination as follows:
(1.16)Sp*(A,B)={f(z)∈Ap:zf′(z)f(z)≺p+Az1+Bz(z∈U)},Kp(A,B)={f(z)∈Ap:1+zf′′(z)f′(z)≺p+Az1+Bz(z∈U)},
where A≠pB, -1≦B≦0. We immediately know that
(1.17)f(z)∈Kp(A,B)iffzf′(z)p∈Sp*(A,B).Then, we have the next theorem.Theorem 1.2.
Iff(z)∈𝒮p*(A,B)(-1<B≦0), then  f(z)∈𝒮b*(p), where
(1.18)b=|B(-pB+Re(A))cos⁡φ+BIm⁡(A)sinφ+|A-pB||1-B2eiφ(0≦φ<2π).
Espesially, f(z)∈𝒮p*(A,-1) if and only if f(z)∈𝒮b*(p) where b=(p+A)/2.Proof.
Supposing thatzf′(z)/f(z)≺(p+Az)/(1-z), it follows from Remark 1.1 that
(1.19)Re[(p+A¯)zf′(z)f(z)]>p2-|A|22,
that is, that
(1.20)Re[2(p+A¯)zf′(z)f(z)]>Re[2p(p+A¯)]-|p+A|2.
This means that
(1.21)Re[2(p+A¯)|p+A|2(zf′(z)f(z)-p)]>-1,
which implies that
(1.22)Re[1(1/2)(p+A)(zf′(z)f(z)-p)]>-1.
Therefore, f(z)∈𝒮b*, where b=(p+A)/2. The converse is also completed.
Next, for the case-1<B≦0, by the definition of the class 𝒮b*(A,B), if a tangent line l2 of the circle C2 containing the point p is parallel to the straight line L:(cosθ)u+(sinθ)v=0(-π≦θ∃<π), and the image F(𝕌) by F(z)=zf′(z)/f(z) is covered by the circle C2, then there exists a non-zero complex number b with arg(b)=θ+π and |b|=d(l2,p) such that f(z)∈𝒮b*(p), where d(l2,p) is the distance between the tangent line l2 and the point p. Now, for the function p(z)=(p+Az)/(1+Bz)(A≠pB,-1<B≦0), the image p(𝕌) is equivalent to
(1.23)C2={ω∈C:|ω-p-AB1-B2|<|A-pB|1-B2},
and the point ξ on ∂C2={ω∈ℂ:|ω-(p-AB)/(1-B2)|=|A-pB|/(1-B2)} can be written by
(1.24)ξ:=ξ(θ)=|A-pB|1-B2eiθ+p-AB1-B2(-π≦θ∃<π).
Further, the tangent line l2 of the circle C2 through each point ξ(θ) is parallel to the straight line L:(cosθ)u+(sinθ)v=0. Namely, l2 can be represented by
(1.25)l2:(cos⁡θ)(u-|A-pB|cos⁡θ+p-BRe(A)1-B2)+(sinθ)(v-|A-pB|sinθ-BIm⁡(A)1-B2)=0,
which implies that
(1.26)l2:(cos⁡θ)u+(sinθ)v-|A-pB|+{p-BRe(A)}cos⁡θ-BIm⁡(A)sinθ1-B2=0.
Then, we see that the distance d(l2,p) between the point p and the above tangent line l2 of the circle C2 is
(1.27)|cos⁡θ×p+sinθ×0-|A-pB|+{p-BRe(A)}cos⁡θ-BIm⁡(A)sinθ1-B2|=|-B(-pB+Re(A))cos⁡θ-BIm⁡(A)sinθ+|A-pB||1-B2.
Therefore, if the subordination
(1.28)zf′(z)f(z)≺p+Az1+Bz(A≠pB,-1<B≦0)
holds true, then f(z)∈𝒮b* where
(1.29)b=|-B(-pB+Re(A))cos⁡θ-BIm⁡(A)sinθ+|A-pB||1-B2ei(θ+π).
Finally, setting φ=θ+π(0≦φ<2π), the proof of the theorem is completed.Noonan and Thomas [8, 9] have stated the qth Hankel determinant as (1.30)Hq(n)=det⁡(anan+1⋯an+q-1an+1an+2⋯an+q⋮⋮⋱⋮an+q-1an+q⋯an+2q-2)(n,q∈N={1,2,3,…}).
This determinant is discussed by several authors with q=2. For example, we can know that the functional |H2(1)|=|a3-a22| is known as the Fekete-Szegö problem, and they consider the further generalized functional |a3-μa22|, where a1=1 and μ is some real number (see, [1]). The purpose of this investigation is to find the sharp upper bounds of the functional |ap+2-μap+12| for functions f(z)∈𝒮p*(A,B) or 𝒦p(A,B).


## 2. Preliminary Results

We need some lemmas to establish our results. Applying the Schwarz lemma or subordination principle.Lemma 2.1.
If a functionw(z)∈𝒲, then
(2.1)|w1|≦1.
Equality is attained for w(z)=eiθz for any θ∈ℝ.The following lemma is obtained by applying the Schwarz-Pick lemma (see, e.g., [7]).Lemma 2.2.
For any functionsw(z)∈𝒲,
(2.2)|w2|≦1-|w1|2
holds true. Namely, this gives us the following representation:
(2.3)w2=(1-|w1|2)ζ,
for some ζ(|ζ|≦1).


## 3.p-Valently Janowski Starlike Functions

Our first main result is contained inTheorem 3.1.
Iff(z)∈𝒮p*(A,B), then
(3.1)|ap+2-μap+12|≦{|(A-pB){(1-2μ)A-((p+1)-2pμ)B}|2(|(1-2μ)A-((p+1)-2pμ)B|≧1),|A-pB|2(|(1-2μ)A-((p+1)-2pμ)B|≦1),
with equality for
(3.2)f(z)={zp(1+Bz)(pB-A)/BorzpeAz(B=0)(|(1-2μ)A-((p+1)-2pμ)B|≧1),zp(1+Bz2)(pB-A)/2Borzpe(A/2)z2(B=0)(|(1-2μ)A-((p+1)-2pμ)B|≦1).Proof.
Letf(z)∈𝒮p*(A,B). Then, there exists the function w(z)∈𝒲 such that
(3.3)zf′(z)f(z)=p+Aw(z)1+Bw(z),
which means that
(3.4)(n-p)an=∑k=pn-1(A-kB)akwn-k(n≧p+1),
where ap=1. Thus, by the help of the relation in Lemma 2.2, we see that
(3.5)|ap+2-μap+12|=|12(A-pB){w2+(A-(p+1)B)w12}-μ(A-pB)2w12|=|A-pB|2|(1-w12)ζ+{(A-(p+1)B)-2μ(A-pB)}w12|.
Then, by Lemma 2.1, supposing that 0≦w1≦1 without loss of generality, and applying the triangle inequality, it follows that
(3.6)|(1-w12)ζ+{(A-(p+1)B)-2μ(A-pB)}w12|≦1+{|(A-(p+1)B)-2μ(A-pB)|-1}w12≦{|(A-(p+1)B)-2μ(A-pB)|(|(A-(p+1)B)-2μ(A-pB)|≧1;w1=1),1(|(A-(p+1)B)-2μ(A-pB)|≦1;w1=0).Especially, takingμ=(p+1)/2p in Theorem 3.1, we obtain the following corollary.Corollary 3.2.
Iff(z)∈𝒮p*(A,B), then
(3.7)|ap+2-p+12pap+12|≦{|A(A-pB)|2p(|A|≧p),|A-pB|2(|A|≦p),
with equality for
(3.8)f(z)={zp(1+Bz)(pB-A)/BorzpeAz(B=0)(|A|≧p),zp(1+Bz2)(pB-A)/2Borzpe(A/2)z2(B=0)(|A|≦p).Furthermore, puttingA=p-2α and B=-1 for some α(0≦α<p) in Theorem 3.1, we arrive at the following result by Hayami and Owa [2, Theorem  3].Corollary 3.3.
Iff(z)∈𝒮p*(α), then
(3.9)|ap+2-μap+12|≦{(p-α){(2(p-α)+1)-4(p-α)μ}(μ≦12),p-α(12≦μ≦p-α+12(p-α)),(p-α){4(p-α)μ-(2(p-α)+1)}(μ≧p-α+12(p-α)),
with equality for
(3.10)f(z)={z(1-z)2(p-α)(μ≦12orμ≧p-α+12(p-α)),z(1-z2)p-α(12≦μ≦p-α+12(p-α)).


## 4.p-Valently Janowski Convex Functions

Similarly, we consider the functional|ap+2-μap+12| for p-valently Janowski convex functions.Theorem 4.1.
Iff(z)∈𝒦p(A,B), then
(4.1)|ap+2-μap+12|≦{p|(A-pB){((p+1)2-2p(p+2)μ)A-((p+1)3-2p2(p+2)μ)B}|2(p+1)2(p+2),(|((p+1)2-2p(p+2)μ)A-((p+1)3-2p2(p+2)μ)B|≧(p+1)2),p|A-pB|2(p+2),(|((p+1)2-2p(p+2)μ)A-((p+1)3-2p2(p+2)μ)B|≦(p+1)2),
with equality for
(4.2)f(z)={zp2F1(p,p-AB;p+1;-Bz)orzp1F1(p,p+1;Az)(B=0),(|((p+1)2-2p(p+2)μ)A-((p+1)3-2p2(p+2)μ)B|≧(p+1)2),zp2F1(p2,pB-A2B;1+p2;-Bz2)orzp1F1(p2,1+p2;A2z2)(B=0),(|((p+1)2-2p(p+2)μ)A-((p+1)3-2p2(p+2)μ)B|≦(p+1)2),
where 2F1(a,b;c;z) represents the ordinary hypergeometric function and 1F1(a,b;z) represents the confluent hypergeometric function.Proof.
By the help of the relation (1.17) and Theorem 3.1, if f(z)∈𝒦p(A,B), then
(4.3)|p+2pap+2-μ(p+1)2p2ap+12|=p+2p|ap+2-(p+1)2p(p+2)μap+12|,≦C(μ),
where C(μ) is one of the values in Theorem 3.1. Then, dividing the both sides by (p+2)/p and replacing ((p+1)2/p(p+2))μ by μ, we obtain the theorem.Now, lettingμ=(p+1)3/2p2(p+2) in Theorem 4.1, we have the following corollary.Corollary 4.2.
Iff(z)∈𝒦p(A,B), then
(4.4)|ap+2-(p+1)32p2(p+2)ap+12|≦{|A(A-pB)|2(p+2)(|A|≧p),p|A-pB|2(p+2)(|A|≦p),
wiht equality for
(4.5)f(z)={zp2F1(p,p-AB;p+1;-Bz)orzp1F1(p,p+1;Az)(B=0)(|A|≧p),zp2F1(p2,pB-A2B;1+p2;-Bz2)orzp1F1(p2,1+p2;A2z2)(B=0)(|A|≦p),
where 2F1(a,b;c;z) represents the ordinary hypergeometric function and 1F1(a,b;z) represents the confluent hypergeometric function.Moreover, we suppose thatA=p-2α and B=-1 for some α(0≦α<p). Then, we arrive at the result by Hayami and Owa [2, Theorem  4].Corollary 4.3.
Iff(z)∈𝒦p(α), then
(4.6)|ap+2-μap+12|≦{p(p-α){(p+1)2(2(p-α)+1)-4p(p+2)(p-α)μ}(p+1)2(p+2)(μ≦(p+1)22p(p+2)),p(p-α)p+2((p+1)22p(p+2)≦μ≦(p+1)2(p-α+1)2p(p+2)(p-α)),p(p-α){4p(p+2)(p-α)μ-(p+1)2(2(p-α)+1)}(p+1)2(p+2)(μ≧(p+1)2(p-α+1)2p(p+2)(p-α)),
with equality for
(4.7)f(z)={zp2F1(p,2(p-α);p+1;z)(μ≦(p+1)22p(p+2)orμ≧(p+1)2(p-α+1)2p(p+2)(p-α)),zp2F1(p2,p-α;1+p2;z2)((p+1)22p(p+2)≦μ≦(p+1)2(p-α+1)2p(p+2)(p-α)),
where 2F1(a,b;c;z) represents the ordinary hypergeometric function.



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