ON CERTAIN ANALYTIC UNIVALENT FUNCTIONSON CERTAIN ANALYTIC UNIVALENT FUNCTIONS 309...
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IJMMS 25:5 (2001) 305–310 PII. S0161171201004781 http://ijmms.hindawi.com © 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 functions B(α) to investigate some properties for this class. The angular estimates of functions in the class B(α) are obtained. Finally, we derive some interesting conditions for the class of strongly starlike and strongly convex of order α in the open unit disk. 2000 Mathematics Subject Classification. Primary 30C45. 1. Introduction. Let A denote the class of functions of the form f (z) = z + ∞ n=2 a n z n , (1.1) which are analytic in the open unit disk U ={z : |z| < 1}. A function f (z) belonging to A is said to be starlike of order α if it satisfies Re zf (z) f (z) >α (z ∈ U) (1.2) for some α(0 ≤ α< 1). We denote by S ∗ α the subclass of A consisting of functions which are starlike of order α in U . Also, a function f (z) belonging to A is said to be convex of order α if it satisfies Re 1 + zf (z) f (z) >α (z ∈ U) (1.3) for some α(0 ≤ α< 1). We denote by C α the subclass of A consisting of functions which are convex of order α in U . If f (z) ∈ A satisfies arg zf (z) f (z) < π 2 α (z ∈ U) (1.4) for some α(0 ≤ α< 1), then f (z) said to be strongly starlike of order α in U , and this class denoted by ¯ S ∗ α . If f (z) ∈ A satisfies arg 1 + zf (z) f (z) < π 2 α (z ∈ U) (1.5) for some α(0 ≤ α< 1), then we say that f (z) is strongly convex of order α in U , and we denote by ¯ C α the class of all such functions.
Transcript of ON CERTAIN ANALYTIC UNIVALENT FUNCTIONSON CERTAIN ANALYTIC UNIVALENT FUNCTIONS 309...
IJMMS 25:5 (2001) 305–310PII. S0161171201004781http://ijmms.hindawi.com© 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 functions B(α) to investigate some propertiesfor this class. The angular estimates of functions in the class B(α) are obtained. Finally, wederive some interesting conditions for the class of strongly starlike and strongly convexof order α in the open unit disk.
2000 Mathematics Subject Classification. Primary 30C45.
1. Introduction. Let A denote the class of functions of the form
f(z)= z+∞∑n=2
anzn, (1.1)
which are analytic in the open unit disk U = {z : |z| < 1}. A function f(z) belongingto A is said to be starlike of order α if it satisfies
Re
(zf ′(z)f(z)
)>α (z ∈U) (1.2)
for some α (0 ≤ α < 1). We denote by S∗α the subclass of A consisting of functionswhich are starlike of order α in U . Also, a function f(z) belonging to A is said to beconvex of order α if it satisfies
Re
(1+ zf
′′(z)f ′(z)
)>α (z ∈U) (1.3)
for some α (0 ≤ α < 1). We denote by Cα the subclass of A consisting of functionswhich are convex of order α in U .
If f(z) ∈ A satisfies ∣∣∣∣∣arg(zf ′(z)f(z)
)∣∣∣∣∣< π2 α (z ∈U) (1.4)
for some α (0 ≤ α < 1), then f(z) said to be strongly starlike of order α in U , andthis class denoted by S̄∗α .
If f(z)∈A satisfies ∣∣∣∣∣arg(1+ zf
′′(z)f ′(z)
)∣∣∣∣∣< π2 α (z ∈U) (1.5)
for some α (0≤α< 1), then we say that f(z) is strongly convex of order α in U , andwe denote by C̄α the class of all such functions.
306 B. A. FRASIN AND M. DARUS
The object of the present paper is to investigate various properties of the followingclass of analytic functions defined as follows.
Definition 1.1. A function f(z) ∈ A is said to be a member of the class B(α) ifand only if ∣∣∣∣∣z
2f ′(z)f 2(z)
−1
∣∣∣∣∣< 1−α (1.6)
for some α (0≤α< 1) and for all z ∈U .Note that condition (1.6) implies
Re
(z2f ′(z)f 2(z)
)>α. (1.7)
2. Main results. In order to derive our main results, we have to recall here thefollowing lemmas.
Lemma 2.1 (see [2]). Let f(z)∈A satisfy the condition
∣∣∣∣∣z2f ′(z)f 2(z)
−1
∣∣∣∣∣< 1 (z ∈U), (2.1)
then f is univalent in U .
Lemma 2.2 (see [1]). Let w(z) be analytic in U and such that w(0) = 0. Then if|w(z)| attains its maximum value on circle |z| = r < 1 at a point z0 ∈U , we have
z0w′(z0)= kw(z0), (2.2)
where k≥ 1 is a real number.
Lemma 2.3 (see [3]). Let a function p(z) be analytic in U, p(0) = 1, and p(z) ≠0 (z ∈U). If there exists a point z0 ∈U such that
∣∣arg(p(z))∣∣< π2α, for |z|< ∣∣z0∣∣, ∣∣arg(p(z0))∣∣= π2 α, (2.3)
with 0<α≤ 1, then we have
z0p′(z0)
p(z0) = ikα, (2.4)
where
k≥ 12
(a+ 1
a
)≥ 1 when arg
(p(z0))= π
2α,
k≤−12
(a+ 1
a
)≤−1 when arg
(p(z0))=−π
2α,
p(z0)1/α =±ai, (a > 0).
(2.5)
We begin with the statement and the proof of the following result.
ON CERTAIN ANALYTIC UNIVALENT FUNCTIONS 307
Theorem 2.4. If f(z)∈A satisfies
∣∣∣∣∣(zf(z)
)′′f ′(z)
− 2zf ′(z)f(z)
∣∣∣∣∣< 1−α2−α (z ∈U), (2.6)
for some α (0≤α< 1), then f(z)∈ B(α).Proof. We define the function w(z) by
z2f ′(z)f 2(z)
= 1+(1−α)w(z). (2.7)
Then w(z) is analytic in U and w(0) = 0. By the logarithmic differentiations, we getfrom (2.7) that (
zf(z))′′
f ′(z)− 2zf ′(z)
f(z)= (1−α)zw′(z)
1+(1−α)w(z) . (2.8)
Suppose there exists z0 ∈U such that
max|z|<|z0|
∣∣w(z)∣∣= ∣∣w(z0)∣∣= 1, (2.9)
then from Lemma 2.2, we have (2.2).
Letting w(z0)= eiθ , from (2.8), we have
∣∣∣∣∣(z0f
(z0))′′
f ′(z0) − 2z0f ′
(z0)
f(z0)
∣∣∣∣∣=∣∣∣∣∣ (1−α)keiθ1+(1−α)eiθ
∣∣∣∣∣≥ 1−α2−α, (2.10)
which contradicts our assumption (2.6). Therefore |w(z)|< 1 holds for all z ∈ U . Wefinally have ∣∣∣∣∣z
2f ′(z)f 2(z)
−1
∣∣∣∣∣= (1−α)∣∣w(z)∣∣< 1−α (z ∈U), (2.11)
that is, f(z)∈ B(α).Taking α= 0 in Theorem 2.4 and using Lemma 2.1 we have the following corollary.
Corollary 2.5. If f(z)∈A satisfies
∣∣∣∣∣(zf(z)
)′′f ′(z)
− 2zf ′(z)f(z)
∣∣∣∣∣< 12
(z ∈U), (2.12)
then f is univalent in U .
Next, we prove the following theorem.
Theorem 2.6. Let f(z)∈A. If f(z)∈ B(α), then∣∣∣∣arg
(f(z)z
)∣∣∣∣< π2 α (z ∈U), (2.13)
for some α (0<α< 1) and (2/π)tan−1α−α= 1.
308 B. A. FRASIN AND M. DARUS
Proof. We define the function p(z) by
f(z)z
= p(z)= 1+∞∑n=2
anzn−1. (2.14)
Then we see that p(z) is analytic in U, p(0) = 1, and p(z) ≠ 0 (z ∈ U). It followsfrom (2.14) that
z2f ′(z)f 2(z)
= 1p(z)
(1+ zp
′(z)p(z)
). (2.15)
Suppose there exists a point z0 ∈U such that
∣∣arg(p(z))∣∣< π2α, for |z|< ∣∣z0∣∣, ∣∣arg(p(z0))∣∣= π2 α. (2.16)
Then, applying Lemma 2.3, we can write that
z0p′(z0)
p(z0) = ikα, (2.17)
where
k≥ 12
(a+ 1
a
)≥ 1 when arg
(p(z0))= π
2α,
k≤−12
(a+ 1
a
)≤−1 when arg
(p(z0))=−π
2α,
p(z0)1/α =±ai, (a > 0).
(2.18)
Therefore, if arg(p(z0))=πα/2, then
z20f ′(z0)
f 2(z0) = 1
p(z0)(1+ z0p
′(z0)p(z0))= a−αe−iπα/2(1+ikα). (2.19)
This implies that
arg
(z20f ′
(z0)
f 2(z0))= arg
(1
p(z0)(1+ z0p
′(z0)p(z0)))
=−π2α+arg(1+iαk)≥−π
2α+tan−1α
= π2
(2π
tan−1α−α)= π
2
(2.20)
if2π
tan−1α−α= 1. (2.21)
Also, if arg(p(z0))=−πα/2, we have
arg
(z20f ′
(z0)
f 2(z0))≤−π
2(2.22)
if2π
tan−1α−α= 1. (2.23)
These contradict the assumption of the theorem.
ON CERTAIN ANALYTIC UNIVALENT FUNCTIONS 309
Thus, the function p(z) has to satisfy
∣∣arg(p(z))∣∣< π2α (z ∈U) (2.24)
or ∣∣∣∣arg(f(z)z
)∣∣∣∣< π2 α (z ∈U). (2.25)
This completes the proof.
Now, we prove the following theorem.
Theorem 2.7. Let p(z) be analytic in U, p(z)≠ 0 in U and suppose that∣∣∣∣∣arg
(p(z)+ z
3f ′(z)f 2(z)
p′(z))∣∣∣∣∣< π2 α (z ∈U), (2.26)
where 0<α< 1 and f(z)∈ B(α), then we have
∣∣arg(p(z))∣∣< π2α (z ∈U). (2.27)
Proof. Suppose there exists a point z0 ∈U such that
∣∣arg(p(z))∣∣< π2α, for |z|< ∣∣z0∣∣, ∣∣arg(p(z0))∣∣= π2 α. (2.28)
Then, applying Lemma 2.3, we can write that
z0p′(z0)
p(z0) = ikα, (2.29)
where
k≥ 12
(a+ 1
a
)when arg
(p(z0))= π
2α,
k≤−12
(a+ 1
a
)when arg
(p(z0))=−π
2α,
p(z0)1/α =±ai, (a > 0).
(2.30)
Then it follows that
arg
(p(z0)+ z30f ′
(z0)
f 2(z0) p′(z0)
)= arg
(p(z0)(
1+ z20f ′
(z0)
f 2(z0) zp′
(z0)
p(z0)))
= arg
(p(z0)(
1+iz20f ′
(z0)
f 2(z0) αk
)).
(2.31)
When arg(p(z0))=πα/2, we have
arg
(p(z0)+ z30f ′
(z0)
f 2(z0) p′(z0)
)= arg
(p(z0))+arg
(1+iz
20f ′
(z0)
f 2(z0) αk
)>π2α, (2.32)
because
Rez20f ′
(z0)
f 2(z0) αk> 0 and therefore arg
(1+iz
20f ′
(z0)
f 2(z0) αk
)> 0. (2.33)
310 B. A. FRASIN AND M. DARUS
Similarly, if arg(p(z0))=−πα/2, then we obtain that
arg
(p(z0)+ z30f ′
(z0)
f 2(z0) p′(z0)
)= arg
(p(z0))+arg
(1+iz
20f ′
(z0)
f 2(z0) αk
)<−π
2α, (2.34)
because
Rez20f ′
(z0)
f 2(z0) αk< 0 and therefore arg
(1+iz
20f ′
(z0)
f 2(z0) αk
)< 0. (2.35)
Thus we see that (2.32) and (2.34) contradict our condition (2.26). Consequently, weconclude that ∣∣arg(p(z))∣∣< π
2α (z ∈U). (2.36)
Taking p(z)= zf ′(z)/f(z) in Theorem 2.7, we have the following corollary.
Corollary 2.8. If f(z)∈A satisfying
∣∣∣∣∣arg(zf ′(z)f(z)
+ z3f ′(z)f 3(z)
((zf ′(z)
)′ − z(f ′(z)
)2f(z)
))∣∣∣∣∣< π2 α (z ∈U), (2.37)
where 0<α< 1 and f(z)∈ B(α), then f(z)∈ S̄∗α .Taking p(z)= 1+zf ′′(z)/f ′(z) in Theorem 2.7, we have the following corollary.
Corollary 2.9. If f(z)∈A satisfying
∣∣∣∣∣arg((zf ′(z)
)′f ′(z)
+ z3
f 3(z)
((zf ′′(z)
)′ − z(f ′′(z)
)2f ′(z)
))∣∣∣∣∣< π2 α, (2.38)
where 0<α< 1 and f(z)∈ B(α), then f(z)∈ C̄α .
References
[1] I. S. Jack, Functions starlike and convex of order α, 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 andM. 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, Bangi 43600 Selangor, Malaysia
M. Darus: School of Mathematical Sciences, Faculty of Sciences and Technology,UKM, Bangi 43600 Selangor, Malaysia
E-mail address: [email protected]
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