The Fekete Szegö Problem for Valently Janowski Starlike and Convex Functions

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Volume 2011, Article ID 583972,11pages doi:10.1155/2011/583972

Research Article

The Fekete-Szeg ¨o Problem for

p

-Valently Janowski

Starlike and Convex Functions

Toshio Hayami

1

_{and Shigeyoshi Owa}

2

1_{School of Science and Technology, Kwansei Gakuin University, Sanda, Hyogo 669-1337, Japan}

2_{Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan}

Correspondence should be addressed to Shigeyoshi Owa,owa@math.kindai.ac.jp

Received 13 January 2011; Accepted 4 May 2011

Academic Editor: A. Zayed

Copyrightq2011 T. Hayami and S. 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.

Forp-valently Janowski starlike and convex functions defined by applying subordination for the

generalized Janowski function, the sharp upper bounds of a functional|ap2−μa2_p1|related to the

Fekete-Szeg ¨o problem are given.

1. Introduction

LetApdenote the family of functionsfznormalized by

fz zp ∞

np1

anzn _p_1,_2,_{3, . . .}_, _1.1

which are analytic in the open unit diskU{z∈C:|z|<1}. Furtheremore, letWbe the class

of functionswzof the form

wz ∞

k1

wkzk, 1.2

which are analytic and satisfy|wz|<1 inU. Then, a functionwz∈ Wis called the Schwarz

function. Iffz∈ Apsatisfies the following condition

Re

11

b

zfz

fz −p

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for some complex number bb /0, then fz is said to bep-valently starlike function of

complex orderb. We denote byS_b∗pthe subclass ofApconsisting of all functionsfzwhich

arep-valently starlike functions of complex orderb. Similarly, we say thatfzis a member

of the classKbpofp-valently convex functions of complex orderbinUiffz∈ Apsatisfies

Re

11

b

zfz

fz −

p−1>0 z∈U, 1.4

for some complex numberbb /0.

Next, letFz zfz/fz uiv andb ρeiϕ _{ρ >} _0, ₀ _{ϕ <} _{2π. Then, the}

condition of the definition ofS_b∗pis equivalent to

Re

1 1

b

zfz

fz −p

1cosϕ

ρ

u−psinϕ

ρ v >0. 1.5

We denote bydl1, pthe distance between the boundary linel1 :cosϕu sinϕv

ρ−pcosϕ0 of the half plane satisfying the condition1.5and the pointF0 p. A simple

computation gives us that

dl1, p cosϕ×psinϕ×0ρ−pcosϕ

cos2_ϕ_sin2_ϕ ρ, 1.6

that is, thatdl1, pis always equal to|b| ρregardless ofϕ. Thus, if we consider the circle

C1with center atpand radiusρ, then we can know the definition ofS_b∗pmeans thatFUis

covered by the half plane separated by a tangent line ofC1and containingC1. Forp1, the

same things are discussed by Hayami and Owa3.

Then, we introduce the following function:

pz 1Az

1Bz −1B < A1, 1.7

which has been investigated by Janowski4. Therefore, the functionpzgiven by1.7is

said to be the Janowski function. Furthermore, as a generalization of the Janowski function,

Kuroki et al.6have investigated the Janowski function for some complex parametersAand

Bwhich satisfy one of the following conditions:

i A /B, |B|<1, |A|1, Re1−AB|A−B|;

ii A /B, |B|1, |A|1, 1−AB >0.

1.8

Here, we note that the Janowski function generalized by the conditions1.8is analytic and

univalent inU, and satisfies Repz>0z∈U. Moreover, Kuroki and Owa5discussed

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the conditions forAandBto satisfy Repz>0. In the present paper, we consider the more

general Janowski functionpzas follows:

pz pAz

1Bz

p1,2,3, . . ., 1.9

for some complex parameterAand some real parameterBA /pB, −1 B0. Then, we

don’t need to discuss the other cases because for the function:

qz pA1z

1B1z

A1, B1∈C, A1/pB1, |B1|1, 1.10

lettingB1|B1|eiθ_{and replacing}_z_by_−e−iθ_z_in_{1.10, we see that}

pz q−e−iθz p−A1e

−iθ_z

1− |B1|z ≡

pAz

1Bz

A−A1e−iθ, B−|B1|, 1.11

mapsUonto the same circular domain asqU.

Remark 1.1. For the caseB −1 in1.9, we know thatpzmapsUonto the following half plane:

RepApz> p

2_{− |A|}2

2 , 1.12

and for the case−1< B0 in1.9,pzmapsUonto the circular domain

pz−pAB

1−B2

< ApB

1−B2 . 1.13

Letpzandqzbe analytic inU. Then, we say that the functionpzis subordinate

toqzinU, written by

pz≺qz z∈U, 1.14

if there exists a functionwz∈ Wsuch thatpz qwz z∈U. In particular, ifqzis

univalent inU, thenpz≺qzif and only if

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We next define the subclasses ofApby applying the subordination as follows:

S∗

pA, B

fz∈ Ap: zf

_z

fz ≺

pAz

1Bz z∈U

,

KpA, B fz∈ Ap: 1zf

_z

fz ≺

pAz

1Bz z∈U

,

1.16

where_{A /}pB,−1B0. We immediately know that

fz∈ KpA, B iﬀ zfz

p ∈ S

∗

pA, B. 1.17

Then, we have the next theorem.

Theorem 1.2. Iffz∈ S∗

pA, B −1< B0, thenfz∈ Sb∗p, where

b B

−pBReAcosϕBImAsinϕ A−pB

1−B2 e

iϕ ₀_{ϕ <}_2π_. _1.18

Espesially,fz∈ S_p∗A,−1if and only iffz∈ S_b∗pwhereb pA/2.

Proof. Supposing thatzfz/fz≺pAz/1−z, it follows from Remark1.1that

RepAzf

_z

fz

> p

2_{− |A|}2

2 , 1.19

that is, that

Re

2pAzf

_z

fz

>Re2ppA− pA 2. 1.20

This means that

Re ⎡ ⎢

⎣2

pA

_p_A 2

zfz

fz −p

⎤

⎥

⎦>−1, 1.21

which implies that

Re

1

1/2pA

zfz

fz −p

>−1. 1.22

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Next, for the case−1 < B0, by the definition of the classS∗_bA, B, if a tangent line

l2of the circleC2containing the pointpis parallel to the straight lineL:cosθu sinθv

0−π ∃_{θ < π}, and the imageFUbyFz zfz/fzis covered by the circleC2, then

there exists a non-zero complex numberbwith argb θπ and |b| dl2, psuch that

fz∈ S∗

bp, wheredl2, pis the distance between the tangent linel2and the pointp. Now,

for the functionpz pAz/1Bz _{A /}pB, −1< B0, the imagepUis equivalent

to

C2

ω∈C: ω−p−AB

1−B2

< A−pB

1−B2

, 1.23

and the pointξon∂C2{ω∈C:|ω−p−AB/1−B2_|_|A₋_pB|/1₋_B2_}_{can be written}

by

ξ:ξθ A−pB

1−B2 e

iθp−AB

1−B2

−π ∃

θ < π

. 1.24

Further, the tangent linel2of the circleC2through each pointξθis parallel to the straight

lineL:cosθu sinθv0. Namely,l2can be represented by

l2:cosθ

u− A−pB cosθp−BReA

1−B2

sinθ

v− A−pB sinθ−BImA

1−B2

0,

1.25

which implies that

l2:cosθu sinθv− A−pB

p−BReAcosθ−BImAsinθ

1−B2 0. 1.26

Then, we see that the distancedl2, pbetween the pointpand the above tangent linel2of the

circleC2is

cosθ×psinθ×0− A−pB

p−BReAcosθ−BImAsinθ

1−B2

−B

−pBReAcosθ−BImAsinθ A−pB

1−B2 .

1.27

Therefore, if the subordination

zfz

fz ≺

pAz

1Bz

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holds true, thenfz∈ S∗_bwhere

b −B

−pBReAcosθ−BImAsinθ A−pB

1−B2 eiθπ. 1.29

Finally, settingϕθπ 0ϕ <2π, the proof of the theorem is completed.

Noonan and Thomas8,9have stated theqth Hankel determinant as

Hqn det

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

an an1 · · · anq−1

an1 an2 · · · anq

..

. ... . .. ...

anq−1 anq · · · an2q−2

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

n, q∈N{1,2,3, . . .}. 1.30

This determinant is discussed by several authors withq2. For example, we can know that

the functional|H21| |a3−a2

2|is known as the Fekete-Szeg ¨o problem, and they consider

the further generalized functional|a3−μa2

2|, wherea1 1 and μis some real numbersee,

1. The purpose of this investigation is to find the sharp upper bounds of the functional

|ap2−μa2

p1|for functionsfz∈ Sp∗A, BorKpA, B.

2. Preliminary Results

We need some lemmas to establish our results. Applying the Schwarz lemma or subordina-tion principle.

Lemma 2.1. If a functionwz∈ W, then

|w1|1. 2.1

Equality is attained forwz eiθ_z_{for any}_θ_∈_R_.

The following lemma is obtained by applying the Schwarz-Pick lemmasee, e.g.,7.

Lemma 2.2. For any functionswz∈ W,

|w2|1− |w1|2 2.2

holds true. Namely, this gives us the following representation:

w21− |w1|2ζ, 2.3

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3. p

-Valently Janowski Starlike Functions

Our first main result is contained in

Theorem 3.1. Iffz∈ S∗

pA, B, then

ap2−μa2

p1

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

_A₋_pB

1−2μA−p1−2pμB

2 1−2μ

A−p1−2pμB 1,

_A₋_pB

2 1−2μ

A−p1−2pμB 1,

3.1

with equality for

fz

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

zp

1BzpB−A/B orz

p_eAz_B₀ ₁₋_2μ_A₋_p₁₋_2pμ_B ₁_,

zp

1Bz2pB−A/2B orz

p_eA/2z2

B0 1−2μA−p1−2pμB 1.

3.2

Proof. Letfz∈ S_p∗A, B. Then, there exists the functionwz∈ Wsuch that

zfz

fz

pAwz

1Bwz, 3.3

which means that

n−pan

n−1

kp

A−kBakwn−k np1, 3.4

whereap1. Thus, by the help of the relation in Lemma2.2, we see that

ap2−μa2_p1 1

2

A−pB)w2A−p1Bw₁2*−μA−pB2w₁2

A−pB

2

1−w₁2ζA−p1B−2μA−pBw2₁ .

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Then, by Lemma2.1, supposing that 0w11 without loss of generality, and applying the triangle inequality, it follows that

1−w₁2ζA−p1B−2μA−pBw2₁

1 A−p1B−2μA−pB −1w2₁

⎧ ⎨ ⎩

_A₋

p1B−2μA−pB A−p1B−2μA−pB 1; w11,

1 A−p1B−2μA−pB 1; w10.

3.6

Especially, takingμ p1/2pin Theorem3.1, we obtain the following corollary.

Corollary 3.2. Iffz∈ S∗

pA, B, then

ap2−p1

2p a

2 p1

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

_A

A−pB

2p

|A|p,

_A₋_pB

2

|A|p,

3.7

with equality for

fz ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

zp

1BzpB−A/B or z

p_eAz_B₀ _|A|_p_,

zp

1Bz2pB−A/2B or z

p_eA/2z2

B0 |A|p.

3.8

Furthermore, puttingA p−2αandB −1 for someα0 α < pin Theorem3.1,

we arrive at the following result by Hayami and Owa2, Theorem 3.

Corollary 3.3. Iffz∈ S∗

pα, then

ap2−μa2

p1

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

p−α2p−α1−4p−αμ μ 1

2

,

p−α

1

2 μ

p−α1

2p−α

,

p−α4p−αμ−2p−α1

μ p−α1

2p−α

,

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with equality for fz ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z

1−z2p−α

μ1

2 orμ

p−α1

2p−α

,

z

1−z2p−α

1

2 μ

p−α1

2p−α

.

3.10

4. p

-Valently Janowski Convex Functions

Similarly, we consider the functional|ap2−μa2

p1|forp-valently Janowski convex functions.

Theorem 4.1. Iffz∈ KpA, B, then

ap2−μa2_p1

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

p A−pB)p12−2pp2μA−p13−2p2_p₂_μ_B*

2p12p2 ,

p12−2pp2μA−p13−2p2_p₂_μ_B _p₁2_,

p A−pB

2p2 ,

p12−2pp2μA−p13−2p2_p₂_μ_B _p₁2_,

4.1

with equality for

fz ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

zp_2F1

p, p− A

B;p1;−Bz

orzp_1F1_{p, p}_1;_Az_B_0,

p12−2pp2μA−p13−2p2_p₂_μ_B _p₁2_,

zp_2F1

p

2,

pB−A

2B ; 1

p

2;−Bz

2

orzp_1F1

p

2,1

p 2; A 2z 2

B0,

p12−2pp2μA−p13−2p2_p₂_μ_B _p₁2_,

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where2F1a, b;c;zrepresents the ordinary hypergeometric function and1F1a, b;zrepresents the confluent hypergeometric function.

Proof. By the help of the relation1.17and Theorem3.1, iffz∈ KpA, B, then

p2

p ap2−μ

p12

p2 a

2 p1

p2

p

ap2−

p12

pp2μa

2 p1

,

Cμ,

4.3

whereCμis one of the values in Theorem3.1. Then, dividing the both sides byp2/p

and replacingp12/pp2μbyμ, we obtain the theorem.

Now, lettingμ p13/2p2p2in Theorem4.1, we have the following corollary.

Corollary 4.2. Iffz∈ KpA, B, then

ap2−

p13

2p2_p₂a

2 p1

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

_A

A−pB

2p2

|A|p,

p A−pB

2p2

|A|p,

4.4

wiht equality for

fz ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

zp_2F1

p, p−A

B;p1;−Bz

orzp_1F1_{p, p}_1;_Az_B₀ _|A|_p_,

zp_2F1

p

2,

pB−A

2B ; 1

p

2;−Bz

2

or zp_1F1

p

2,1

p

2;

A

2z

2

B0 |A|p,

4.5

where2F1a, b;c;zrepresents the ordinary hypergeometric function and1F1a, b;zrepresents the confluent hypergeometric function.

Moreover, we suppose thatAp−2αandB −1 for someα0 α < p. Then, we

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Corollary 4.3. Iffz∈ Kpα, then

ap2−μa2_p1

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

pp−α)p122p−α1−4pp2p−αμ*

p12p2

μ

p12

2pp2

,

pp−α

p2

p12

2pp2 μ

p12p−α1

2pp2p−α

,

pp−α)4pp2p−αμ−p122p−α1*

p12p2

μ

p12p−α1

2pp2p−α

,

4.6

with equality for

fz ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

zp_2F1_p,₂_p₋_α_;_p_1;_z

μ

p12

2pp2 or μ

p12p−α1

2pp2p−α

,

zp_2F1p

2, p−α; 1

p

2;z

2 p1

2

2pp2 μ

p12p−α1

2pp2p−α

,

4.7

where2F1a, b;c;zrepresents the ordinary hypergeometric function.

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Mathematical Society, vol. 8, no. 2, pp. 85–89, 1933.

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General Mathematics, vol. 17, no. 4, pp. 29–44, 2009.

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4 W. Janowski, “Extremal problems for a family of functions with positive real part and for some related

families,” Annales Polonici Mathematici, vol. 23, pp. 159–177, 1970.

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Inequalities in Pure and Applied Mathematics, vol. 10, no. 2, Article ID 36, pp. 1–11, 2009.

6 K. Kuroki, S. Owa, and H. M. Srivastava, “Some subordination criteria for analytic functions,” Bulletin

de la Société des Sciences et des Lettres deŁód´z, vol. 52, pp. 27–36, 2007.

7 Z. Nehari, Conformal Mapping, McGraw-Hill, New York, NY, USA, 1952.

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Proceedings of the London Mathematical Society, vol. 25, pp. 503–524, 1972.

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