On certain analytic univalent functions

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ON CERTAIN ANALYTIC UNIVALENT FUNCTIONS

B. A. FRASIN and M. DARUS

(Received 10 March 2000)

Abstract.We consider the class of analytic functionsB(α)to investigate some properties for this class. The angular estimates of functions in the classB(α)are obtained. Finally, we derive some interesting conditions for the class of strongly starlike and strongly convex of orderαin the open unit disk.

2000 Mathematics Subject Classification. Primary 30C45.

1. Introduction. LetAdenote the class of functions of the form

f (z)=z+

n=2

anzn, (1.1)

which are analytic in the open unit diskU= {z:|z|<1}. A functionf (z)belonging toAis said to be starlike of orderαif it satisfies

Re

zf(z)

f (z)

> α (z∈U) (1.2)

for someα (0≤α <1). We denote byS∗

α the subclass ofAconsisting of functions which are starlike of orderαinU. Also, a functionf (z)belonging toAis said to be convex of orderαif it satisfies

Re

1+zff(z)(z)

> α (z∈U) (1.3)

for someα (0≤α <1). We denote by the subclass ofAconsisting of functions which are convex of orderαinU.

Iff (z) Asatisfies arg

zf(z)

f (z)

2α (z∈U) (1.4)

for someα (0≤α <1), thenf (z)said to be strongly starlike of orderαinU, and this class denoted by ¯S∗

α. Iff (z)∈Asatisfies

arg

1+zf(z)

f(z)

2α (z∈U) (1.5)

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The object of the present paper is to investigate various properties of the following class of analytic functions defined as follows.

Definition1.1. A functionf (z)∈Ais said to be a member of the classB(α)if and only if

z

2f(z)

f2(z) 1

<1−α (1.6)

for someα (0≤α <1)and for allz∈U. Note that condition (1.6) implies

Re

z2f(z)

f2(z)

> α. (1.7)

2. Main results. In order to derive our main results, we have to recall here the following lemmas.

Lemma2.1(see [2]). Letf (z)∈Asatisfy the condition

z

2f(z)

f2(z) 1

<1 (z∈U), (2.1)

thenf is univalent inU.

Lemma2.2 (see [1]). Let w(z)be analytic inU and such thatw(0)=0. Then if

|w(z)|attains its maximum value on circle|z| =r <1at a pointz0∈U, we have

z0wz0=kwz0, (2.2)

wherek≥1is a real number.

Lemma2.3(see [3]). Let a functionp(z)be analytic in U, p(0)=1, andp(z)≠ 0(z∈U). If there exists a pointz0∈Usuch that

arg

p(z)<π2α, for|z|<z0, argpz0=π2α, (2.3)

with0< α≤1, then we have

z0pz0

pz0 =ikα, (2.4)

where

k≥12

a+1a

1 whenargpz0=π2α,

k≤ −12

a+a1

≤ −1 when argpz0= −π2α,

pz01/α= ±ai, (a >0).

(2.5)

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Theorem2.4. Iff (z)∈Asatisfies

zf (z) f(z)

2zf(z)

f (z)

<12αα (z∈U), (2.6)

for someα (0≤α <1), thenf (z)∈B(α). Proof. We define the functionw(z)by

z2f(z)

f2(z) =1+(1−α)w(z). (2.7)

Thenw(z)is analyticinUandw(0)=0.By the logarithmicdifferentiations, we get

from (2.7) that

zf (z) f(z)

2zf(z)

f (z) =

(1−α)zw(z)

1+(1−α)w(z). (2.8)

Suppose there existsz0∈Usuch that

max |z|<|z0|

w(z)=w

z0=1, (2.9)

then from Lemma 2.2, we have (2.2). Lettingw(z0)=eiθ, from (2.8), we have

z0fz0 fz0

2z0fz0

fz0 =

(1−α)ke 1+(1−α)eiθ

12αα, (2.10)

which contradicts our assumption (2.6). Therefore|w(z)|<1 holds for allz∈U. We finally have

z

2f(z)

f2(z) 1

=(1−α)w(z)<1−α (z∈U), (2.11)

that is,f (z)∈B(α).

Takingα=0 in Theorem 2.4 and using Lemma 2.1 we have the following corollary. Corollary2.5. Iff (z)∈Asatisfies

zf (z) f(z)

2zf(z)

f (z)

<12 (z∈U), (2.12)

thenf is univalent inU.

Next, we prove the following theorem.

Theorem2.6. Letf (z)∈A. Iff (z)∈B(α), then

argf (z) z

2α (z∈U), (2.13)

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Proof. We define the functionp(z)by

f (z)

z =p(z)=1+

n=2

anzn−1. (2.14)

Then we see thatp(z)is analyticinU, p(0)=1, andp(z)≠0(z∈U). It follows from (2.14) that

z2f(z)

f2(z) = 1 p(z)

1+zpp(z)(z)

. (2.15)

Suppose there exists a pointz0∈Usuch that arg

p(z)<π

2α, for|z|<z0, arg

pz0=π

2α. (2.16) Then, applying Lemma 2.3, we can write that

z0pz0

pz0 =ikα, (2.17)

where

k≥1

2

a+1

a

1 when argpz0=π

2α, k≤ −12

a+a1

≤ −1 when argpz0= −π2α,

pz01/α= ±ai, (a >0).

(2.18)

Therefore, if arg(p(z0))=πα/2, then

z2 0f

z0 f2z0 =

1 pz0

1+z0p

z0 pz0

=a−αe−iπα/2(1+ikα). (2.19)

This implies that

arg

z2 0f

z0 f2z0

=arg

1 pz0

1+z0p

z0 pz0

= −π2α+arg(1+iαk)≥ −π2α+tan1α

2

2

πtan1α−α

2

(2.20)

if

2

πtan1α−α=1. (2.21)

Also, if arg(p(z0))= −πα/2, we have

arg

z2 0f

z0 f2z0

≤ −π2 (2.22)

if

2

πtan1α−α=1. (2.23)

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Thus, the functionp(z)has to satisfy arg

p(z)<π2α (z∈U) (2.24)

or

arg f (z)

z

2α (z∈U). (2.25)

This completes the proof.

Now, we prove the following theorem.

Theorem2.7. Letp(z)be analytic inU, p(z)≠0inUand suppose that

arg

p(z)+zf3f2(z)(z)p(z)

2α (z∈U), (2.26)

where0< α <1andf (z)∈B(α), then we have arg

p(z)<π2α (z∈U). (2.27)

Proof. Suppose there exists a pointz0∈Usuch that arg

p(z)<π2α, for|z|<z0, argpz0=π2α. (2.28)

Then, applying Lemma 2.3, we can write that

z0pz0

pz0 =ikα, (2.29)

where

k≥12

a+a1

when argpz0=π2α,

k≤ −12

a+a1

when argpz0= −π2α,

pz01/α= ±ai, (a >0).

(2.30)

Then it follows that

arg

pz0+z30f

z0 f2z0 p

z0

=arg

pz0

1+z0f2

z0 f2z0

zpz0

pz0

=arg

pz0

1+iz20f

z0 f2z0 αk

.

(2.31)

When arg(p(z0))=πα/2, we have

arg

pz0+z30f

z0 f2z0 p

z0

=argpz0+arg

1+iz20f

z0 f2z0 αk

2α, (2.32)

because

Rez20f

z0

f2z0 αk >0 and therefore arg

1+iz20f

z0 f2z0 αk

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Similarly, if arg(p(z0))= −πα/2, then we obtain that

arg

pz0+z03f

z0 f2z0 p

z0

=argpz0+arg

1+iz02f

z0 f2z0 αk

<−π2α, (2.34)

because

Rez20f

z0

f2z0 αk <0 and therefore arg

1+iz20f

z0 f2z0 αk

<0. (2.35)

Thus we see that (2.32) and (2.34) contradict our condition (2.26). Consequently, we conclude that

arg

p(z)<π2α (z∈U). (2.36)

Takingp(z)=zf(z)/f (z)in Theorem 2.7, we have the following corollary.

Corollary2.8. Iff (z)∈Asatisfying

arg

zf(z)

f (z) +

z3f(z)

f3(z)

zf(z)z

f(z)2

f (z)

2α (z∈U), (2.37)

where0< α <1andf (z)∈B(α), thenf (z)∈S¯ α.

Takingp(z)=1+zf(z)/f(z)in Theorem 2.7, we have the following corollary.

Corollary2.9. Iff (z)∈Asatisfying

arg zf

(z)

f(z) +

z3 f3(z)

zf(z)zf(z)2

f(z)

2α, (2.38)

where0< α <1andf (z)∈B(α), thenf (z)∈C¯α.

References

[1] I. S. Jack,Functions starlike and convex of orderα, J. London Math. Soc. (2)3(1971), 469– 474. MR 43#7611. Zbl 224.30026.

[2] M. Nunokawa,On some angular estimates of analytic functions, Math. Japon.41(1995), no. 2, 447–452. MR 96c:30013. Zbl 822.30014.

[3] S. Ozaki and M. Nunokawa,The Schwarzian derivative and univalent functions, Proc. Amer. Math. Soc.33(1972), 392–394. MR 45#8821. Zbl 233.30011.

B. A. Frasin: School of Mathematical Sciences, Faculty of Sciences and Technology, UKM, Bangi43600Selangor, Malaysia

M. Darus: School of Mathematical Sciences, Faculty of Sciences and Technology, UKM, Bangi43600Selangor, Malaysia

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