© Hindawi Publishing Corp.
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 byCα the subclass ofAconsisting of functions which are convex of orderαinU.
Iff (z) ∈ Asatisfies arg
zf(z)
f (z)
<π2α (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)
<π2α (z∈U) (1.5)
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)
Theorem2.4. Iff (z)∈Asatisfies
zf (z) f(z) −
2zf(z)
f (z)
<12−−αα (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 iθ 1+(1−α)eiθ
≥12−−αα, (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
<π
2α (z∈U), (2.13)
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+tan−1α
=π2
2
πtan−1α−α
=π2
(2.20)
if
2
πtan−1α−α=1. (2.21)
Also, if arg(p(z0))= −πα/2, we have
arg
z2 0f
z0 f2z0
≤ −π2 (2.22)
if
2
πtan−1α−α=1. (2.23)
Thus, the functionp(z)has to satisfy arg
p(z)<π2α (z∈U) (2.24)
or
arg f (z)
z
<π2α (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)
<π2α (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
>π2α, (2.32)
because
Rez20f
z0
f2z0 αk >0 and therefore arg
1+iz20f
z0 f2z0 αk
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)
<π2α (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)
<π2α, (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
