IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation58397210.1155/2011/583972583972Research ArticleThe Fekete-Szegö Problem for p-Valently Janowski Starlike and Convex FunctionsHayamiToshio1OwaShigeyoshi2ZayedA.1School of Science and TechnologyKwansei Gakuin UniversitySanda, Hyogo 669-1337Japankwansei.ac.jp2Department of MathematicsKinki UniversityHigashi-OsakaOsaka 577-8502Japankindai.ac.jp20111472011201113012011040520112011Copyright © 2011 Toshio Hayami and Shigeyoshi Owa.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1. Introduction

Let 𝒜p denote the family of functions f(z) normalized by f(z)=zp+n=p+1anzn(p=1,2,3,), which are analytic in the open unit disk 𝕌={z:|z|<1}. Furtheremore, let 𝒲 be the class of functions w(z) of the form w(z)=k=1wkzk, 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 Re[1+1b(zf(z)f(z)-p)]>0(zU), for some complex number b  (b0), 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 Re[1+1b(zf′′(z)f(z)-(p-1))]>0(zU), for some complex number b  (b0).

Next, let F(z)=zf(z)/f(z)=u+iv and b=ρeiφ  (ρ>0,  0φ<2π). Then, the condition of the definition of 𝒮b*(p) is equivalent to Re[1+1b(zf(z)f(z)-p)]=1+cosφρ(u-p)+sinφρv>0.

We denote by d(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 d(l1,p)=|cosφ×p+sinφ×0+ρ-pcosφ|cos2φ+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 .

Then, we introduce the following function: p(z)=1+Az1+Bz(-1B<A1), which has been investigated by Janowski . 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.  have investigated the Janowski function for some complex parameters A and B which satisfy one of the following conditions: (i)AB,|B|<1,|A|1,Re(1-AB¯)|A-B|;(ii)AB,|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  (z𝕌). Moreover, Kuroki and Owa  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: p(z)=p+Az1+Bz(p=1,2,3,), for some complex parameter A and some real parameter B  (ApB,  -1B0). Then, we don't need to discuss the other cases because for the function: q(z)=p+A1z1+B1z(A1,B1C,  A1pB1,  |B1|1), letting B1=|B1|eiθ and replacing z by -e-iθz in (1.10), we see that p(z)=q(-e-iθz)=p-A1e-iθz1-|B1|zp+Az1+Bz(A=-A1e-iθ,  B=-|B1|), maps 𝕌 onto the same circular domain as q(𝕌).

Remark 1.1.

For the case B=-1 in (1.9), we know that p(z) maps 𝕌 onto the following half plane: Re(p+A¯)p(z)>p2-|A|22, and for the case -1<B0 in (1.9), p(z) maps 𝕌 onto the circular domain |p(z)-p+AB1-B2|<|A+pB|1-B2.

Let p(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 p(z)q(z)(zU), if there exists a function w(z)𝒲 such that p(z)=q(w(z))(z𝕌). In particular, if q(z) is univalent in 𝕌, then p(z)q(z) if and only if p(0)=q(0),p(U)q(U).

We next define the subclasses of 𝒜p by applying the subordination as follows: Sp*(A,B)={f(z)Ap:zf(z)f(z)p+Az1+Bz  (zU)},Kp(A,B)={f(z)Ap:1+zf′′(z)f(z)p+Az1+Bz  (zU)}, where ApB, -1B0. We immediately know that f(z)Kp(A,B)iff  zf(z)pSp*(A,B).

Then, we have the next theorem.

Theorem 1.2.

If f(z)𝒮p*(A,B)  (-1<B0), then f(z)𝒮b*(p), where b=|B(-pB+Re(A))cosφ+BIm(A)sinφ+|A-pB||1-B2eiφ(0φ<2π). Espesially, f(z)𝒮p*(A,-1) if and only if f(z)𝒮b*(p) where b=(p+A)/2.

Proof.

Supposing that zf(z)/f(z)(p+Az)/(1-z), it follows from Remark 1.1 that Re[(p+A¯)zf(z)f(z)]>p2-|A|22, that is, that Re[2(p+A¯)zf(z)f(z)]>Re[2p(p+A¯)]-|p+A|2. This means that Re[2(p+A¯)|p+A|2(zf(z)f(z)-p)]>-1, which implies that 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<B0, 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  (-πθ<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)  (ApB,  -1<B0), the image p(𝕌) is equivalent to C2={ωC:|ω-p-AB1-B2|<|A-pB|1-B2}, and the point ξ on C2={ω:|ω-(p-AB)/(1-B2)|=|A-pB|/(1-B2)} can be written by ξ:=ξ(θ)=|A-pB|1-B2eiθ+p-AB1-B2(-πθ<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 l2:(cosθ)(u-|A-pB|cosθ+p-BRe(A)1-B2)+(sinθ)(v-|A-pB|sinθ-BIm(A)1-B2)=0, which implies that l2:(cosθ)u+(sinθ)v-|A-pB|+{p-BRe(A)}cosθ-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 |cosθ×p+sinθ×0-|A-pB|+{p-BRe(A)}cosθ-BIm(A)sinθ1-B2|=|-B(-pB+Re(A))cosθ-BIm(A)sinθ+|A-pB||1-B2. Therefore, if the subordination zf(z)f(z)p+Az1+Bz(ApB,  -1<B0) holds true, then f(z)𝒮b* where b=|-B(-pB+Re(A))cosθ-BIm(A)sinθ+|A-pB||1-B2ei(θ+π). Finally, setting φ=θ+π  (0φ<2π), the proof of the theorem is completed.

Noonan and Thomas [8, 9] have stated the qth Hankel determinant as Hq(n)=det(anan+1an+q-1an+1an+2an+qan+q-1an+qan+2q-2)(n,qN={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, ). 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 function w(z)𝒲, then |w1|1. Equality is attained for w(z)=eiθz for any θ.

The following lemma is obtained by applying the Schwarz-Pick lemma (see, e.g., ).

Lemma 2.2.

For any functions w(z)𝒲, |w2|1-|w1|2 holds true. Namely, this gives us the following representation: w2=(1-|w1|2)ζ, for some ζ  (|ζ|1).

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M170"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math></inline-formula>-Valently Janowski Starlike Functions

Our first main result is contained in

Theorem 3.1.

If f(z)𝒮p*(A,B), then |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 f(z)={zp(1+Bz)(pB-A)/B  or  zpeAz(B=0)(|(1-2μ)A-((p+1)-2pμ)B|1),zp(1+Bz2)(pB-A)/2B  or  zpe(A/2)z2(B=0)(|(1-2μ)A-((p+1)-2pμ)B|1).

Proof.

Let f(z)𝒮p*(A,B). Then, there exists the function w(z)𝒲 such that zf(z)f(z)=p+Aw(z)1+Bw(z), which means that (n-p)an=k=pn-1(A-kB)akwn-k(np+1), where ap=1. Thus, by the help of the relation in Lemma 2.2, we see that |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 0w11 without loss of generality, and applying the triangle inequality, it follows that |(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, taking μ=(p+1)/2p in Theorem 3.1, we obtain the following corollary.

Corollary 3.2.

If f(z)𝒮p*(A,B), then |ap+2-p+12pap+12|{|A(A-pB)|2p(|A|p),|A-pB|2(|A|p), with equality for f(z)={zp(1+Bz)(pB-A)/B  or  zpeAz(B=0)(|A|p),zp(1+Bz2)(pB-A)/2B  or  zpe(A/2)z2  (B=0)(|A|p).

Furthermore, putting A=p-2α and B=-1 for some α  (0α<p) in Theorem 3.1, we arrive at the following result by Hayami and Owa [2, Theorem  3].

Corollary 3.3.

If f(z)𝒮p*(α), then |ap+2-μap+12|{(p-α){(2(p-α)+1)-4(p-α)μ}(μ12),p-α(12μp-α+12(p-α)),(p-α){4(p-α)μ-(2(p-α)+1)}(μp-α+12(p-α)), with equality for f(z)={z(1-z)2(p-α)(μ12  or  μp-α+12(p-α)),z(1-z2)p-α(12μp-α+12(p-α)).

4. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M192"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math></inline-formula>-Valently Janowski Convex Functions

Similarly, we consider the functional |ap+2-μap+12| for p-valently Janowski convex functions.

Theorem 4.1.

If f(z)𝒦p(A,B), then |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 f(z)={zp2F1(p,p-AB;p+1;-Bz)  or  zp1F1(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)  or  zp1F1(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 |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, letting μ=(p+1)3/2p2(p+2) in Theorem 4.1, we have the following corollary.

Corollary 4.2.

If f(z)𝒦p(A,B), then |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 f(z)={zp2F1(p,p-AB;p+1;-Bz)  or  zp1F1(p,p+1;Az)  (B=0)(|A|p),zp2F1(p2,pB-A2B;1+p2;-Bz2) or  zp1F1(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 that A=p-2α and B=-1 for some α(0α<p). Then, we arrive at the result by Hayami and Owa [2, Theorem  4].

Corollary 4.3.

If f(z)𝒦p(α), then |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 f(z)={zp2F1(p,2(p-α);p+1;z)(μ(p+1)22p(p+2)  or  μ(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.

FeketeM.SzegöG.Eine bemerkung uber ungerade schlichte funktionenJournal of the London Mathematical Society193382858910.1112/jlms/s1-8.2.85HayamiT.OwaS.Hankel determinant for p-valently starlike and convex functions of order αGeneral Mathematics200917429442733349HayamiT.OwaS.New properties for starlike and convex functions of complex orderInternational Journal of Mathematical Analysis201041–439622657758ZBL1197.30006JanowskiW.Extremal problems for a family of functions with positive real part and for some related familiesAnnales Polonici Mathematici1970231591770267103KurokiK.OwaS.Some subordination criteria concerning the Sălăgean operatorJournal of Inequalities in Pure and Applied Mathematics2009102111362511929ZBL1167.26321KurokiK.OwaS.SrivastavaH. M.Some subordination criteria for analytic functionsBulletin de la Société des Sciences et des Lettres de Łódź20075227362427011ZBL1162.30312NehariZ.Conformal Mapping1952New York, NY, USAMcGraw-Hill0045823NoonanJ. W.ThomasD. K.On the hankel determinants of areally mean p-valent functionsProceedings of the London Mathematical Society197225503524030647910.1112/plms/s3-25.3.503NoonanJ. W.ThomasD. K.On the second hankel determinant of areally mean p-valent functionsTransactions of the American Mathematical Society197622323373460422607