Radius of strongly starlikeness for certain analytic functions

http://ijmms.hindawi.com © Hindawi Publishing Corp.

RADIUS OF STRONGLY STARLIKENESS FOR CERTAIN

ANALYTIC FUNCTIONS

OH SANG KWON and SHIGEYOSHI OWA

Received 13 April 2001

For analytic functionsf (z)=zp_+ap

+1zp+1+···in the open unit diskUand a polynomial Q(z)of degreen >0, the functionF(z)=f (z)[Q(z)]β/n_{is introduced. The object of the} present paper is to determine the radius ofp-valently strongly starlikeness of orderγfor F(z).

2000 Mathematics Subject Classification: 30C45.

1. Introduction. LetᏭp(pis a fixed integer1) denote the class of functionsf (z) of the form

f (z)=zp₊ ∞ k=p+1

akzk (1.1)

which are analytic in the open unit diskU= {z∈C:|z|<1}. LetΩdenote the class of bounded functions w(z)analytic inU and satisfying the conditionsw(0)=0 and |w(z)| |z|, z ∈U. We use ᏼ to denote the class of functions p(z)=1+c1z+

c2z2+ ···which are analytic inUand satisfy Rep(z) >0(z∈U). For 0α < pand|λ|< π/2, we denote by᏿λ

p(α), the family of functionsg(z)∈Ꮽp which satisfy

zg_(z)

g(z) ≺

p+2(p−α)e−iλcosλ−pz

1−z , z∈U, (1.2)

where≺means the subordination. From the definition of subordinations, it follows thatg(z)∈Ꮽphas the representation

zg_(z)

g(z) =

p+2(p−α)e−iλ_{cosλ−pw(z)}

1−w(z) , (1.3)

wherew(z)∈Ω. Clearly,᏿λ

p(α)is a subclass ofp-valentλ-spiral functions of order

α. Forλ=0, we have the class᏿∗_p(α), 0α < p, ofp-valent starlike functions of orderα, investigated by Goluzina [5].

A functionf (z)∈Ꮽpis said to bep-valently strongly starlike of orderγ, 0< γ1, if it satisfies

arg

_zf(z)

f (z)

Ba¸sgöze [1,2] has obtained sharp inequalities of univalence (starlikeness) for certain polynomials of the formF(z)=f (z)[Q(z)]β/n_{, whereβis real andQ(z)is a} polyno-mial of degreen >0 all of whose zeros are outside or on the unit circle{z:|z| =1}. Rajasekaran [7] extended Ba¸sgöze’s results for certain classes of analytic functions of the formF(z)=f (z)[Q(z)]β/n_{. Recently, Patel [6] generalized some of the work of} Rajasekaran and Ba¸sgöze for functions belonging to the class᏿λ

p(α). That is, deter-mine the radius of starlikeness for some classes ofp-valent analytic functions of the polynomial formF(z).

In the present paper, we extend the results of Patel [6]. Thus, we determine the radius ofp-valently strongly starlike of orderγfor polynomials of the formF(z)in such problems.

2. Some lemmas. Before proving our next results, we need the following lemmas.

Lemma2.1(see Gangadharan [4]). For|z| r <1,|zk| =R > r,

_z−z_zk+_R2r₋2_r2

_R2Rr_−r2. (2.1)

Lemma2.2(see Ratti [8]). Ifφ(z)is analytic inUand|φ(z)| 1forz∈U, then for |z| =r <1,

zφ₁₊(z)_zφ(z)+φ(z)₁₋1_r. (2.2)

Lemma2.3(see Causey and Merkes [3]). Ifp(z)=1+c1z+c2z+··· ∈ᏼ, then for |z| =r <1,

zp_p(z)(z)₁2₋r_r2. (2.3)

This estimate is sharp.

Lemma2.4(see Patel [6]). Supposeg(z)∈᏿λ

p(α). Then for|z| =r <1,

zg_g(z)(z)−p+2(p−α)eiλr2cosλ 1−r2

2(p−α)rcosλ

1−r2 . (2.4)

This result is sharp.

Lemma2.5(see Gangadharan [4]). IfRaRe(a)sin((π/2)γ)−Im(a)cos((π/2)γ), Im(a)0, then the disk|w−a| Ra is contained in the sector |argw| (π/2)γ, 0< γ1.

3. Main results. Our first theorem is the following one.

Theorem3.1. Suppose that

F(z)=f (z) Q(z)β/n, (3.1)

R1. Iff (z)∈Ꮽpsatisfies

Re _{f (z)}

g(z)

1/δ

>0, 0< δ1, z∈U, (3.2)

Re _g(z)

h(z)

>0, z∈U, (3.3)

for someg(z)∈Ꮽp and h(z)∈᏿λp(α), thenF(z) is p-valently strongly starlike of orderγin|z|< R(γ), whereR(γ)is the smallest positive root of the equation

r4_(p+β)sinπ 2γ

+2(p−α)cosλsin

λ−π 2γ

+r3 _{|β|R+2(p−α)cosλ+2(δ+1)}

−r2_p1+R2_+βsinπ 2γ

+2(p−α)R2_{cosλsinλ−}π 2γ

−r |β|R+2(p−α)R2_{cosλ+2(δ+1)R}2_+pR2_sinπ 2γ

=0.

(3.4)

Proof. We choose a suitable branch of(f (z)/g(z))1/δ_{so that(f (z)/g(z))}1/δ_is analytic inUand takes the value 1 atz=0. Thus from (3.2) and (3.3), we have

F(z)=pδ

1(z)p2h(z) Q(z) β/n

, (3.5)

wherepj(z)∈ᏼ(j=1,2). Then

zF_(z)

F(z) =δ zp

1(z)

p1(z) +

zp

2(z)

p2(z) +

zh_(z)

h(z) + β n

n

k=1

z

z−zk. (3.6)

Sinceh(z)∈᏿λ

p(α), byLemma 2.4, we have

zh_h(z)(z)−

p+2(p−α)e₁₋iλ_rr₂2cosλ2(p−₁α)r_−r2cosλ. (3.7)

Using (3.6) and (3.7) with Lemmas2.1and2.3, we get

zF_F(z)(z)−

p+2(p−α)e₁₋iλ_rr22cosλ−

βr2

R2_−r2

2

(p−α)rcosλ+r (δ+1)

1−r2 +

|β|Rr R2_−r2.

(3.8)

Using Lemma 2.5, we get that the above disk is contained in the sector |argw|< (π/2)γprovided the inequality

2(p−α)rcosλ+r (δ+1)

1−r2 +

|β|Rr R2_−r2

p+2(p−α)r2cos2λ 1−r2 −

βr2

R2_−r2

sin _π

2γ

−2(p−α)r2sinλcosλ 1−r2 cos

_π

2γ

is satisfied. The above inequality is simplified toT (r )0, where

T (r )=r4_{p−2(p−α)cos}2_λ+βsinπ 2γ

+(p−α)sin 2λcos _π

2γ

+r3 _{|β|R+2(p−α)cosλ+2(δ+1)}

+r2_−pR2_{−p+2(p−α)R}2_cos2_λ−βsinπ 2γ

−(p−α)R2_{sin 2λcos}π 2γ

−r |β|R+2(p−α)R2_{cosλ+2(δ+1)R}2_+pR2_sinπ 2γ

.

(3.10)

SinceT (0) >0 andT (1) <1, there exists a real root ofT (r )=0 in(0,1). LetR(γ)be the smallest positive root ofT (r )=0 in(0,1). ThenF(z)isp-valent strongly starlike of orderγin|z|< R(γ).

Remark3.2. ForR=1 andγ=1,Theorem 3.1reduces to a result by Patel [6]. Theorem3.3. Suppose thatF(z)is given by (3.1). If f (z)∈Ꮽpsatisfies (3.2) for someg(z)∈᏿λ

p(α), thenF(z)isp-valently strongly starlike of orderγ in|z|< R(γ), whereR(γ)is the smallest positive root of the equation

r4_(p+β)sinπ 2γ

+2(p−α)cosλsin

λ−π 2γ

+r3 _{|β|R+2(p−α)cosλ+2δ}

−r2_p1+R2_+βsinπ 2γ

+2(p−α)R2_{cosλsinλ−}π 2γ

−r |β|R+2(p−α)R2_cosλ+2δR2_+pR2_sinπ 2γ

=0.

(3.11)

Proof. Iff (z)∈Ꮽpsatisfies (3.2) for someg(z)∈᏿λp(α), then

zF_(z)

F(z) =δ zp_(z)

p(z) + zg_(z)

g(z) + β n

n

k=1

z

z−zk. (3.12)

UsingLemma 2.4, we get

zg_g(z)(z)−p+2(p−α)eiλr2cosλ 1−r2

2(p−α)rcosλ

1−r2 . (3.13)

By (3.12) and (3.13) with Lemmas2.1and2.3, we have

zF_F(z)(z)−

p+2(p−α)e₁₋iλ_rr22cosλ−

βr2

R2_−r2

2

(p−α)rcosλ+r δ

1−r2 +

|β|Rr R2_−r2.

(3.14)

The remaining parts of the proof can be proved by a method similar to the one given in the proof ofTheorem 3.1.

Corollary3.4. Suppose thatf (z)is inᏭp. IfRe(f (z)/g(z)) >0forz∈U and

g(z)∈᏿∗

p(α), thenf (z)isp-valently starlike for

|z|< p

(p+1−α)+α2_−2α+2p+1. (3.15)

Theorem3.5. Suppose thatF(z)is given by (3.1). Iff (z)∈Ꮽpsatisfies

f (z)_g(z)1/δ−1<1, 0< δ1, psin _π

2γ

> δ, (3.16)

Re _g(z)

h(z)

>0, z∈U (3.17)

for someg(z)∈Ꮽp and h(z)∈᏿λp(α), thenF(z) is p-valently strongly starlike of orderγin|z|< R(γ), whereR(γ)is the smallest positive root of the equation

r4

(p+β)sin _π

2γ

+2(p−α)cosλsin

λ−π₂γ

+r3 _{|β|R+2(p−α)cosλ+2+δ}

−r2_p1+R2_+βsinπ 2γ

+2(p−α)R2_{cosλsinλ−}π 2γ

+δ

−r |β|R+2(p−α)R2_{cosλ+2(δ+1)R}2_+pR2_{sin π}

2γ

−δR2_=0. (3.18)

Proof. We choose a suitable branch of(f (z)/g(z))1/δ_{so that(f (z)/g(z))}1/δ_is analytic inUand takes the value 1 atz=0. From (3.16), we deduce that

f (z)=g(z)1+w(z)δ, w(z)∈Ω. (3.19)

So that

F(z)=p(z)h(z)1+zφ(z)δ Q(z)β/n, (3.20)

whereφ(z)is analytic inUand satisfies|φ(z)| 1 andp∈ᏼforz∈U. We have

zF_(z)

F(z) = zh_(z)

h(z) + zp_(z)

p(z) +δ

_zφ(z)+φ(z)

1+zφ(z)

+_nβ

n

k=1

z

z−zk. (3.21)

UsingLemma 2.4and (3.21), we have

zF_F(z)(z)−

p+2(p−α)e₁₋iλ_rr22cosλ

2

(p−α)rcosλ+r+δ(1+r )

1−r2 +

|β|Rr R2_−r2.

(3.22)

Theorem3.6. Suppose thatF(z)is given by (3.1). Iff (z)∈Ꮽpsatisfies (3.16) for someg(z)∈᏿λ

p(α), thenF(z)isp-valently strongly starlike of orderγ in|z|< R(γ), whereR(γ)is the smallest positive root of the equation

r4_(p+β)sinπ 2γ

+2(p−α)cosλsin

λ−π 2γ

+r3 _{|β|R+2(p−α)cosλ+δ}

−r2_p1+R2_{+βsin π}

2γ

+2(p−α)R2_cosλsin

λ−π 2γ

+δ

−r |β|R+2(p−α)R2_cosλ+δR2_+pR2_sinπ 2γ

−δR2_=0.

(3.23)

Proof. We choose a suitable branch of(f (z)/g(z))1/δ_{so that(f (z)/g(z))}1/δ_is analytic inUand takes the value 1 atz=0. Sincef (z)∈Ꮽpsatisfies (3.16) for some

g(z)∈᏿λ

p(α), we have

F(z)=g(z)1+zφ(z) Q(z)β/n, (3.24)

whereφ(z)is analytic inU and satisfies the condition|φ(z)| 1 forz∈U. Thus, we have

zF_(z)

F(z) = zg_(z)

g(z) +δ

_zφ(z)+φ(z)

1+zφ(z)

+β_n

n

k=1

z

z−zk. (3.25)

UsingLemma 2.4and (3.25), we get

zF_F(z)(z)−

p+2(p−α)e₁₋iλ_rr₂2cosλ

2(p−α)r₁cos_−rλ₂+δ(1+r )+_R|β₂|₋Rr_r2.

(3.26)

Using Lemma 2.5and (3.26) and a method similar to the one given in the proof of Theorem 3.1, we complete the proof of the theorem.

Remark3.7. Some of the results of Patel [6] can be obtained fromTheorem 3.6by takingR=1 andγ=1.

References

[1] T. Ba¸sgöze,On the univalence of certain classes of analytic functions, J. London Math. Soc. (2)1(1969), 140–144.

[2] ,On the univalence of polynomials, Compositio Math.22(1970), 245–252. [3] W. M. Causey and E. P. Merkes,Radii of starlikeness of certain classes of analytic functions,

J. Math. Anal. Appl.31(1970), 579–586.

[4] A. Gangadharan, V. Ravichandran, and T. N. Shanmugam,Radii of convexity and strong starlikeness for some classes of analytic functions, J. Math. Anal. Appl.211(1997), no. 1, 301–313.

[5] E. C. Goluzina,On the coefficients of a class of functions, regular in a disk and having an integral representation in it, J. Soviet Math.2(1974), 606–617.

[7] S. Rajasekaran,A study on extremal problems for certain classes of univalent analytic func-tions, Ph.D. thesis, Indian Institute of Technology Kanpur, India, 1985.

[8] J. S. Ratti,The radius of univalence of certain analytic functions, Math. Z.107(1968), 241– 248.

Oh Sang Kwon: Department of Mathematics, Kyungsung University, Pusan_608-736, Korea

E-mail address:oskwon@star.kyungsung.ac.kr

Shigeyoshi Owa: Department of Mathematics, Kinki University, Higashi-Osaka, Osaka_577-8502, Japan