Generalized Alpha Close to Convex Functions

International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 830592,9pages

doi:10.1155/2009/830592

Research Article

Generalized Alpha-Close-to-Convex Functions

K. Inayat Noor,

_{Halit Orhan,}

_{and Saima Mustafa}

1_{Department of Mathematics, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan} 2_{Department of Mathematics, Faculty of Science, Ataturk University, 25240 Erzurum, Turkey}

Correspondence should be addressed to K. Inayat Noor,[email protected]

Received 5 January 2009; Revised 26 June 2009; Accepted 13 September 2009

Recommended by Teodor Bulboac˘a

We define the classesGβα, k, γas follows:f ∈ Gβα, k, γif and only if, forz ∈ E {z ∈ C: |z| <1},|arg{1−α2_z2_fz/e−iβ_{φz}| ≤ γπ/2, 0 < γ ≤1;α∈0,1;β∈ −π/2, π/2, where}

φis a function of bounded boundary rotation. Coefficient estimates, an inclusion result, arclength problem, and some other properties of these classes are studied.

Copyrightq2009 K. Inayat Noor et al. 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.

1. Introduction

LetAbe the class of functions of the form:

fz z

∞

n2

anzn, 1.1

which are analytic in the open unit diskE{z∈C:|z|<1}.ByS,K,S∗,andCdenote the subclasses ofAwhich are univalent, close-to-convex, starlike, and convex inE,respectively. Let Vk be the class of functions of bounded boundary rotation. Paate 1 showed that a

functionf,defined by1.1andfz/0, is inVkif and only if, forzreiθ,

2π

0

Re

zfz fz

dθ≤kπ. 1.2

It is geometrically obvious thatk≥2 andV2≡C.

A classTkof analytic functions related with the classVkwas introduced and studied

in2. A functionf ∈Ais inTk,k≥2,if and only if there exists a functiong ∈Vksuch that,

LetPdenote the class of analytic functionspdefined by

pz 1

∞

n1

cnzn 1.3

with Repz>0 forz∈E.

We denote Kγas the class of strongly close-to-convex functions of order γ in the sense of Pommerenke3. A functionf∈Abelongs toKγif and only if there existsg∈S∗

such that|Argzfz/gz| ≤πγ/2,forz∈Eandγ ≥0.

ClearlyK0 C,K1 K,and when 0 ≤ γ < 1,Kγis a subset ofK and hence contains only univalent functions. Forγ >1,f∈Kγcan be of infinite valence; see4.

We now define the following.

Definition 1.1. A functionf ∈ A is said to belong toGβα, k, γ, whereβ is a real number,

α∈C:|α| ≤1,k≥2, andγ∈0,1is called generalized alpha-close-to-convex with argument

βif and only if there existsφ∈Vksuch that

Arg

1−α2z2fz e−iβ_φz

≤

γπ

2 , z∈E. 1.4

In1.4, we choose this branch of argument which equalsβ,|β| < πγ/2,γ ∈ 0,1,when

z 0. We note that the condition |α| ≤ 1 implies thatGβα, k, γ is nonempty. From the

normalization conditionsf0 φ0 1,it follows fromDefinition 1.1that Ree−iβ_{>0 and}

therefore|β|< γπ/2.Also, it follows from1.4that iff∈Gβα, k, γ,thenfz/0 forz∈E.

Condition1.4is equivalent to the followingf ∈Gβα, k, γif and only if there existsp ∈P

such that

1−α2_z2_fz

e−iβ_φz pzcos

β γ −isin

β γ

γ

, φ∈Vk. 1.5

We defineGα, k, γthe class of generalizedα-close-to-convex functions as

Gα, k, γ

|β|<π/2

Gβ

α, k, γ. 1.6

Ifα0 in1.6, then the classG0, k,1is identical with the classTkandGα,2,1is the class

Kof close-to-convex functions. AlsoGβα,2,1in the class of close-to-convex function with

argumentβwas defined by Goodman and Saff 5. For details of special cases ofGβα,2,1

withφz zin1.4, we refer to6. The special case withγ 1 α,k 2,andφz z

2. Main Results

We now prove the main results as follows.

Theorem 2.1. Letα∈0,1.ThenGα, k, γ⊂G0, k, γ1,where

γ1

γ, αγ 2 πarcsin

α2. 2.1

The constantγ1γ, αcannot be smaller.

Proof. We will use an extended version of the method given in8to prove this result. Forα0,the result is obvious. Letf ∈Gα, k, γ.By1.4,1.5, and1.6, then there exists a functionφ∈Vkand a functionp∈P,|β|< π/2 such that

fz e−iβ_φz

pzcosβ/γ−isinβ/γγ

1−α2_z2 , z∈E. 2.2

Letqz pzcosβ/γ−isinβ/γγ,z∈E.Then we have

Arg

fz e−iβ_φz

Argqz−Arg

1−α2z2< π

2

γ 2 π

Arg1−α2z2

. 2.3

We choose in2.3this branch of argument which is equal−βwhenz0.

Since|Arg1−α2_z2_|<arcsinα2_{,z∈E,we have from2.3f ∈G0, k, γ}

1,whereγ1is

given by2.1. The constantγ1γ, αcannot be smaller. Letα∈0,1be fixed. Let us consider

the pointz0∈Cwith|z0|1 and Arg1−α2z2 −arcsinα2.Letφ0∈Vkbe such thatφ0z0

is finite. Then, let

f₀z e

−iβ_φ 0z

1−α2_z2 pzcos

β γ −isin

β γ

γ

, z∈E, β< π

2, 2.4

where

P0z

1εz

1−εz, ε∈C, |ε|<1,

φ₀z 1δ1z

k/2−1

1δ2zk/21

, |δ1||δ2|1.

2.5

Now, forz∈E, Arg

f₀z e−iβ_φ

0z

Arg p0zcos

β γisin

β γ

γ

and Arge−iβ _−β.Sincep

0 maps the unit circle|z| 1 onto imaginary axis, we may choose

ε0,|ε0| 1 such thatε0/1/z0, P0z0 1ε0z0/1−ε0z0/itanβ,p0z0 ai,a >0.This

means thatp0z0is finite and Argp0z0 π/2.Hence

Arg p0zcos

β γisin

β γ

γ

γπ

2 . 2.7

Thus, from2.4and2.6, we have

Arg

f₀z e−iβ_φ

0z

π 2

γ 2 πarcsin

α2_γ 1

π

2. 2.8

Thereforeγ1cannot be smaller.

Forα 1,consider the sequence{zn},zn eiθn,θn ∈ 0, π/4,n ∈ N 1 such that

limn→ ∞zn1.So

lim

n→ ∞Arg

1−z2_n−π

2. 2.9

Letφ∈Vkwithφeiθfinite andθ∈0, π/2.The functionf1defined as

f₁z e

−iβ_φz

1−z2

1z

1−z

cosβ

γ −isin β γ

γ

, z∈E, β< π

2 2.10

belongs toG1, k, γ.Thus, from2.9, it follows that

lim

n→ ∞

Arg

f₁z e−iβ_φz

nlim→ ∞

Arg

1zn

1−zn

cosβ

γ −isin β γ

γ

−Arg1−z2_n1γπ

2. 2.11

This means thatγ11, γ 1γis best possible.

We note that, forγ1,k≥2,we obtain a result proved in8.

Theorem 2.2. Letf∈Gα, k, γ, α∈0,1.Then, for everyγ∈0,1andθ1,θ1with 0≤θ2−θ1≤

2π, one has

_θ2

θ1 Re

1reiθf

_reiθ

freiθ

dθ >− γk

2 −1−R

π, 2.12

where

R 1

π

ψr, θ2−ψr, θ1

,

ψr, θ −Arg1−α2γ2e2iθarctan α

2_r2_{sin 2θ}

1−α2_r2_{cos 2θ}.

Proof. To prove this result, we shall essentially use the similar method given by Kaplan9. Letf ∈Gα, k, γfor fixedα∈0,1.Thenf satisfies the inequality1.4for someβ, |β|< π/2 andφ ∈Vk.Letφ1z φzeiβ,z∈E.Sincefz/0,φ₁z/0 forz ∈E,we can

define, forzreiθ_{,r∈0,1, θis a real number, the following:}

℘r, θ Arg1−α2r2e2iθfreiθ, 2.14

Vr, θ Argφ₁reiθ_{, 2.15}

ψr, θ Arg1−α2r2e2iθreiθfreiθ℘r, θ θ, 2.16

Vr, θ Argreiθφ₁reiθτr, θ θ. 2.17

The functions℘,τ,ψ, andV are continuous and periodic with period 2π. From1.4, we can choose the branches of argument of℘zandτzas

℘r, θ−τr, θ< γπ

2 , γ∈0,1. 2.18

Now, forφ1∈Vk,it is known10that, forθ1< θ2,zreiθ,

θ2

θ1 Re

zφ₁z φ₁z

dθ >− k

2 −1

π. 2.19

From2.16,2.17, and2.19, we have

ψr, θ2−ψr, θ1 ℘r, θ2 θ2−℘r, θ1−θ1

℘r, θ2−τr, θ2

τr, θ2 θ2−τr, θ1−θ1−

℘r, θ1−τr, θ1

> γπ− k

2 −1

π − rk

2 −1

π.

2.20

Moreover, by2.16, we have

d dθψr, θ

d dθArg

1−α2r2e2iθRe

1reiθf

_reiθ

freiθ

and therefore, from2.20

_θ2

θ1 Re

1reiθf

_reiθ

freiθ

dθ

_θ2

θ1

d

dθψr, θdθ−

_θ2

θ1

Arg1α2r2e2iθdθ

>− γk

2 −1

π−ψr, θ1−ψr, θ2

− γk

2 −1−R

π,

2.22

whereψr, θandRare defined by2.13. This completes the proof.

We note that, forγ 1, k 2, α 0,we obtain the necessary condition forf to be close-to-convex inE,proved in9.

Remark 2.3. FromTheorem 2.2, we can interpret some geometrical meaning for the functions inGα, k, γ.For simplicity, let us suppose that the image domain is bounded by an analytic curveΓ. At a point on Γ, the outward drawn normal turns back at most −γk/2−1−

Rπ,whereAis given by2.13. This is a necessary condition for a functionf to belong to

Gα, k, γ. Goodman4showed that iff ∈Kσ,σ ≥0,then, forzreiθ_{, 0≤θ}

1 < θ2≤2π,

θ2

θ1Rezf

_{z/fzdθ >−σπ.}

We note thatf∈Gα, k, γis univalent fork2γ−R≤4,since

Gα, k, γ⊂K γk

2 −1−R

. 2.23

The functions inKγk/2−1−Rneed not even be finitely valent inEfork2γ−R>4.

Remark 2.4. FromTheorem 2.2and11, Lemma 1.3by Pommerenke, it follows thatGα, k, γ

is a linearly invariant family of orderγk/2−R.Therefore, the image ofEunder functions inGα, k, γcontains the schlicht disk|z|<1/k2γ−R.

Theorem 2.5. Letf ∈Gβα, k, γ,γ ∈ 0,1,|β|< π/2, be of the form1.1. Then|a2| ≤k/2

1γ/2|cosβ/γ|.This estimate is best possible, extremal function beingf0zdefined by2.4.

Proof. Letφz z_n∞₂bnzn,pz 1∞n1cnznin1.5.

Now, it is known that, for functions p of positive real part with γ ∈ 0,1, pγ is subordinate to1z/1−zγ.Also|b2| ≤k/2, see1,12. Therefore, from1.5, we have

2a22b2 e−iβcosβ/γ1γ,and this gives us the required result.

Remark 2.6. Let f ∈ Gα, k, γ,for 2 ≤ k ≤ 4−2γ −R,and be given by 1.1. Thenf is univalent inEbyRemark 2.3andw0fz/w0−fzis univalent inEforw0/0,w0/fz.

Now

w0fz

w0−fz

z a2

1

w0

and therefore|a21/w0| ≤2 and so|1/w0| ≥2/4k1γcosβ/γ,on usingTheorem 2.5.

Hence it follows that the image ofEunderf ∈Gα, k, γwith 2≤k2γ−R≤4 contains the schlicht disc|z|<2/4k 1γcosβ/γ.

FromRemark 2.3, and the results proved for the classKσ,σ ≥0 in4, we at once have the following.

Theorem 2.7. Letf ∈Gα, k, γand be given by1.1. LetFσbe defined by

Fσz

1 2σ1

1z

1−z

σ1

−1

z

∞

n2

Anσzn, 2.25

whereσ γk/2−1−R, andRis given by2.13. ClearlyFσ ∈Gα, k, r.

iDenote byLr, fthe length of the image of the circle|z|r underf and byAr, fthe area off|z| ≤r.Then, for 0≤r <1

aLr, f≤Lr, Fσ,

bAr, f≤Ar, Fσ.

iiForzreiθ_{, 0≤r <1,|fz| 1/2σ11z/1−z}σ1_−1. The functionFσ,defined by2.25, shows that this upper bound is sharp.

Theorem 2.8. Letf ∈Gα, k, γ.Then, for 0< r <1,α, r∈0,1,k≥2,

Lr, f≤cα, k, r 1

1−r

k/2γ

, 2.26

wherecα, k, ris a constant depending uponα,k,andγonly.

Proof. Withzreiθ,

Lr, f

2π

0

_zfzdθ

_2π

0

re

−iβ_{φzpzcosβ/γ−isinβ/γ}γ

1−α2_z2

dθ, φ∈Vk, p∈P, z∈E.

2.27

Forφ∈Vk,it is known10that there exists1, s2∈S∗such that

zφz s1z

k/41/2

s2zk/4−1/2

. 2.28

Also, forp∈Pwe have forzreiθ_,

1 2π

_2π

0

pz2dθ ≤ 13r

2

see13. Now, from2.27,2.28, and2.29, we have

Lr, f≤ c1

α, k, γ rk/4−1/2

1 2π

2π

0

|s1z|k/41/22/2−γdθ

2−γ/2

1 2π

2π

0

pz2

γ/2

≤cα, k, γ 1

1−r

k/2γ

,

2.30

where we have used distortion theorems, subordination for the starlike functions, and Holder’s inequality, andcandc1are constants.

Theorem 2.9. Let f ∈ Gα, k, γ and be given by1.1. Then, forα, γ ∈ 0,1, k ≥ 2, one has

ano1nk/2γ−1,n → ∞whereo1is a constant depending only onk,α, andγ.

Proof. Withzreiθ_{,Cauchy’s theorem gives}

nan

1 2πrn

_2π

0

zfze−inθdθ. 2.31

Thus

n|an| ≤

1 2πrn

2π

0

_{zfzdθ 1/2πr}n_{Lr, f. 2.32}

UsingTheorem 2.8and puttingr 1−1/n,we prove this result.

Acknowledgment

The authors would like to thank the referee for thoughtful comments and suggestions.

References

1 V. Paate, “Uber die konforme abbildung von gebieten deren rander von beschrankter drehung sind,” Annales Academiæ Scientiarum Fennicæ A, vol. 33, 77 pages, 1933.

2 K. Inayat Noor, “On a generalization of close-to-convexity,” International Journal of Mathematics and Mathematical Sciences, vol. 6, no. 2, pp. 327–334, 1983.

3 Ch. Pommerenke, “On close-to-convex analytic functions,” Transactions of the American Mathematical Society, vol. 114, pp. 176–186, 1965.

4 A. W. Goodman, “On close-to-convex functions of higher order,” Annales Universitatis Scientiarum Budapestinensis de Rolando E¨ot¨ous Nominatae. Sectio Mathematica, vol. 25, pp. 17–30, 1972.

5 A. W. Goodman and E. B. Saff, “On the definition of a close-to-convex function,” International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 125–132, 1978.

6 A. Lecko, “Some classes of close-to-convex functions,” Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka i Fizyka, vol. 60, no. 9, pp. 62–70, 1989.

7 W. Hengartner and G. Schober, “On Schlicht mappings to domains convex in one direction,” Commentarii Mathematici Helvetici, vol. 45, pp. 303–314, 1970.

9 W. Kaplan, “Close-to-convex schlicht functions,” The Michigan Mathematical Journal, vol. 1, pp. 169– 185, 1952.

10 D. A. Brannan, “On functions of bounded boundary rotation,” Proceedings of the Edinburgh Mathematical Society, vol. 2, pp. 339–347, 1969.

11 Ch. Pommerenke, “Linear-invariante familien analytischer funktionen I,” Mathematische Annalen, vol. 155, no. 2, pp. 108–154, 1964.

12 A. W. Goodman, Univalent Functions, vol. 1-2, Polygonal Publishing House, Washington, DC, USA, 1983.