Bull. Korean Math. Soc. 53 (2016), No. 1, pp. 215–225
http://dx.doi.org/10.4134/BKMS.2016.53.1.215
A CRITERION FOR BOUNDED FUNCTIONS
Mamoru Nunokawa, Shigeyoshi Owa, and Janusz Sokól
Abstract. We consider a sufficient condition for w(z), analytic in |z| <
1, to be bounded in |z| < 1, where w(0) = w ′ (0) = 0. We apply it
to the meromorphic starlike functions. Also, a certain Briot-Bouquet
differential subordination is considered. Moreover, we prove that if p(z)+
zp′ (z)φ(p(z)) ≺ h(z), then p(z) ≺ h(z), where h(z) = [(1 + z)(1 − z)]α ,
under some additional assumptions on φ(z).
1. Introduction
Let H denote the class of functions analytic in the unit disk D = {z ∈ C :
|z| < 1}, and denote by A the class of analytic functions in D and usually
normalized, i.e., A = {f ∈ H : f (0) = 0, f ′ (0) = 1}. We say that the f ∈ H
is subordinate to g ∈ H in the unit disc D, written f ≺ g if and only if there
exists an analytic function w ∈ H such that |w(z)| ≤ |z| and f (z) = g[w(z)]
for z ∈ D. Therefore f ≺ g in D implies f (D) ⊂ g(D). In particular if g is
univalent in D, then the Subordination Principle says that f ≺ g if and only if
f (0) = g(0) and f (|z| < r) ⊂ g(|z| < r), for all r ∈ (0, 1].
Let β, γ be complex numbers and let p, h ∈ H, with h(0) = p(0). The
first-order differential subordination
zp′ (z)
≺ h(z) (z ∈ D)
p(z) +
βp(z) + γ
is called the Briot-Bouquet differential subordination. A lot of the results on
the Briot-Bouquet differential subordination are collected in [5, Ch.3]. It seems
that among contained there cases was not considered the case γ = 0, β = 1
and
α
1+z
2αz
h(z) =
+
,
1−z
1 − z2
where 0 < α < 1. In this work we consider it.
Received January 27, 2015.
2010 Mathematics Subject Classification. Primary 30C45; Secondary 30C80.
Key words and phrases. analytic, meromorphic, convex, starlike, univalent, Nunokawa’s
lemma, Briot-Bouquet, differential subordination.
c
2016
Korean Mathematical Society
215
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MAMORU NUNOKAWA, SHIGEYOSHI OWA, AND JANUSZ SOKÓL
For integer n ≥ 0, denote by Σn the class of meromorphic functions, defined
in U̇ = {z : 0 < |z| < 1}, which are of the form
1
F (z) = + an z n + an+1 z n+1 + · · · .
z
A function F ∈ Σ0 is said to be starlike if it is univalent and the complement
of F (U̇) is starlike with respect to the origin. Denote by Σ∗0 the class of such
functions. If F ∈ Σ0 , then it is well-known that F ∈ Σ∗0 if and only if
zF ′ (z)
>0
Re −
F (z)
for z ∈ U̇. For α < 1, let
zF ′ (z)
Σ∗n,α = F ∈ Σn : Re −
> α, z ∈ U̇ ,
F (z)
the class of meromorphic-starlike functions of order α. For 0 < α ≤ 1, let
zF ′ (z) απ
Σ∗n (α) = F ∈ Σn : arg −
<
,
z
∈
U̇
F (z) 2
the class of meromorphic-strongly starlike functions of order α.
Definition 1 ([5]). We denote by Q the class of functions f that are analytic
and injective on D \ E(f ), where
E(f ) := {ζ : ζ ∈ ∂D and lim f (z) = ∞},
z→ζ
and are such that
f ′ (ζ) 6= 0
(ζ ∈ ∂(D) \ E(f )).
Lemma 1.1 ([5]). Let q ∈ Q with q(0) = a and let
p(z) = a + an z n + · · ·
be analytic in D with
p(z) 6≡ a and n ∈ N = {1, 2, 3, . . .}.
If p is not subordinate to q, then there exist points
z0 = r0 eiθ ∈ D
and ζ0 ∈ ∂D \ E(q),
for which
p(|z| < r0 ) ⊂ q(D),
p(z0 ) = q(ζ0 )
and
z0 p′ (z0 ) = kζ0 q ′ (ζ0 )
for some k ≥ n.
Lemma 1.1 is a generalization of Jack’s lemma [3]. To prove the main results,
we also need the following generalization of Nunokawa’s lemma, [6], [7], see also
[2].
A CRITERION FOR BOUNDED FUNCTIONS
Lemma 1.2 ([9]). Let p(z) be of the form
∞
X
(1)
p(z) = 1 +
an z n ,
am 6= 0,
217
(z ∈ D),
n=m≥1
with p(z) 6= 0 in D. If there exists a point z0 , |z0 | < 1, such that
|arg {p(z)} | < πα/2
in
|z| < |z0 |
and
|arg {p(z0 )} | = πα/2
for some α > 0, then we have
z0 p′ (z0 )
= ikα,
p(z0 )
where
(2)
k ≥ m a2 + 1 /(2a) when arg {p(z0 )} = πα/2
and
(3)
where
k ≤ −m a2 + 1 /(2a)
when
arg {p(z0 )} = −πα/2,
{p(z0 )}1/α = ±ia, a > 0.
2. Main result
Theorem 2.1. Let w(z) be analytic in D with w(0) = w′ (0) = 0 and suppose
that
s
′
zw
(z)
< 1 − Re {z} (z ∈ D).
w(z) −
(4)
w(z) 1 + Re {z}
Then we have
|w(z)| < 1
(z ∈ D).
Proof. If there exists a point z0 , |z0 | < 1, such that
|w(z)| < 1 (|z| < |z0 |)
and
w(z0 ) = eiθ ,
then from Lemma 1.1, we have
z0 w′ (z0 )
= k ≥ 2.
w(z0 )
Then it follows that
2 2
′
′
iθ
w(z0 ) − z0 w (z0 ) = eiθ − z0 w (z0 ) e
iθ
iθ
w(z0 )
e
1+e 2
2
= (cos θ − k/2) + sin θ 1 −
k
2(1 + cos θ)
2
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MAMORU NUNOKAWA, SHIGEYOSHI OWA, AND JANUSZ SOKÓL
= ϕ(k), say.
Then we have
sin2 θ
ϕ (k) = − (cos θ − k/2) −
1 + cos θ
k
− 1 > 0,
=
1 + cos θ
′
k
1−
1 + cos θ
and
ϕ(2) = (cos θ − 1) + 1 −
2
1
1 + cos θ
2 cos2 θ
− 2 cos θ + 1
1 + cos θ
1 − cos θ
=
1 + cos θ
1 − Re{z0 }
=
.
1 + Re{z0 }
2
sin2 θ
=
Therefore, there exists a point z0 , |z0 | < 1, such that
2
′
w(z0 ) − z0 w (z0 ) ≥ 1 − Re{z0 }
w(z0 ) 1 + Re{z0 }
for all k ≥ 2. It contradicts (4) and it completes the proof.
Corollary 2.2. Let
∞
F (z) =
1 X
+
bn z n
z n=1
be analytic in 0 < |z| < 1 and suppose that for 0 < α < α0
s
zF ′′ (z)
1 − Re {z}
−
F ′ (z) − 2 < 1 + Re {z} (z ∈ D).
Then
zF ′ (z)
−
F (z) − 1 < 1
(z ∈ D),
it follows that F (z) is meromorphic-starlike in D.
For another sufficient condition for strongly starlikeness, we refer to the
recent paper [8].
Theorem 2.3. Let p(z) of the form
(5)
p(z) = 1 +
∞
X
n=2
an z n ,
(z ∈ D).
A CRITERION FOR BOUNDED FUNCTIONS
Suppose that α ∈ (0, 1] and
(6)
p(z) +
zp′ (z)
≺
p(z)
1+z
1−z
α
p(z) ≺
1+z
1−z
α
Then we have
+
2αz
1 − z2
(z ∈ D).
(z ∈ D).
Proof. If there exists a point z0 , |z0 | < 1, such that
| arg {p(z)} | < πα/2
(|z| < |z0 |)
and
| arg {p(z0 )} | = πα/2, p(z0 ) = (±ia)α ,
then from Nunokawa’s Lemma 1.2, we have
z0 p′ (z0 )
= ikα,
p(z0 )
where
k≥
a2 + 1
≥ 1, when arg {p(z0 )} = πα/2
2a
and
a2 + 1
≤ −1, when arg {p(z0 )} = −πα/2.
2a
For the case arg {p(z0 )} = απ/2, we have
k≤−
(7)
p(z0 ) +
z0 p′ (z0 )
= (ia)α + iαk.
p(z0 )
Let us put z = eiθ in the right hand side of (6).
α
α
1+z
2αz
iα
i sin θ
+
+
=
(8)
.
2
1−z
1−z
1 − cos θ
sin θ
It is easy to see that it is possible to find θ0 such that for given a > 0
sin θ0
.
a=
1 − cos θ0
Then
α
α(a2 + 1)
,
=
(9)
αk >
2a
sin θ0
and hence from (7), (8) and (9) we get that
p(z0 ) +
z0 p′ (z0 )
p(z0 )
lies outside the image of the unit disc under the function
α
2αz
1+z
+
,
1−z
1 − z2
219
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MAMORU NUNOKAWA, SHIGEYOSHI OWA, AND JANUSZ SOKÓL
which is convex in the direction of the imaginary axis. It contradicts (5). For
the case arg {p(z0 )} = −απ/2 , in the same way as before, we also can obtain
a contradiction (5), which completes the proof.
For α = 1 Theorem 2.3 becomes the result in [5, p. 140]. For 0 < α < 1
Theorem 2.3 is an extension of Theorem 3.2i [5, p. 97].
Corollary 2.4. Let
f (z) = z +
∞
X
an z n
n=1
be analytic in D and suppose that
α
1+z
2αz
zf ′′ (z)
≺
+
1+ ′
f (z)
1−z
1 − z2
(z ∈ D).
Then we have
α
zf ′ (z)
1+z
(z ∈ D),
≺
f (z)
1−z
it follows that f (z) is strongly starlike of order α in D.
Theorem 2.5. Let h(z) = {(1 + z)/(1 − z)}α, α ∈ (0, 1], and p(z) are analytic
in D with h(0) = p(0) = 1. Assume also that φ(p(z)) is analytic in D, moreover
Re {φ(h(z))} ≥ 0 in D. If
(10)
p(z) + zp′ (z)φ(p(z)) ≺ h(z)
(z ∈ D),
then
p(z) ≺ h(z)
Proof. If there exists a point z0 = e
iθ0
(z ∈ D).
, |z0 | < 1, such that
p(z) 6= h(eiθ ) for all θ, 0 ≤ θ < 2π and |z| < |z0 |
and
p(z0 ) = h(eiθ0 ),
then from the hypothesis of the theorem, we have the following picture, for the
case arg {p(z0 )} = απ/2 < 0.
Then from Nunokawa’s Lemma 1.2, we have
z0 p′ (z0 )
= ikα,
p(z0 )
where
απ/2 = arg {p(z0 )}
and
k ≥ 1, when arg {p(z0 )} = πα/2 > 0
while
k ≤ −1, when arg {p(z0 )} = πα/2 < 0.
A CRITERION FOR BOUNDED FUNCTIONS
✻
Im
✡
✡
❏
❏
221
A = {p(z) : |z| < |z0 |}
✡
✡ B = {h(z) : |z| < 1}
✡
✡
✡
B
✡
r1
Re
✲
A
❏
❏r
c ❏
❏
❏❏
Figure 1. c = p(z0 ) = h(eiθ0 ).
For the case arg {p(z0 )} = απ/2 > 0, 1 ≤ k, we have
arg {p(z0 ) + z0 p′ (z0 )φ(p(z0 ))}
z0 p′ (z0 )
φ(p(z0 ))
= arg p(z0 ) 1 +
p(z0 )
iθ0 = arg h(e ) + arg 1 + iαkφ h(eiθ0 )
> arg h(eiθ0 .
This contradicts the hypothesis (10) and for the case arg {p(z0 )} = απ/2 < 0,
k ≤ −1, applying the same method as the above, we have
arg {p(z0 ) + z0 p′ (z0 )φ(p(z0 ))}
= arg h(eiθ0 ) + arg 1 + iαkφ h(eiθ0 )
< arg h(eiθ0 .
This is also a contradiction and therefore, it completes the proof.
Recall here the well known theorem due to Hallenbeck and Ruscheweyh [4].
Theorem A ([4]). Let the function h be analytic and convex univalent in D
with h(0) = a. Let also p(z) = a + bn z n + bn+1 z n+1 + · · · be analytic in D. If
(11)
p(z) +
zp′ (z)
≺ h(z), z ∈ D
c
222
MAMORU NUNOKAWA, SHIGEYOSHI OWA, AND JANUSZ SOKÓL
for Re {c} ≥ 0, c 6= 0, then
p(z) ≺ qn (z) ≺ h(z), z ∈ D,
c
nz c/n
Rz
c/n−1
h(t)dt. Moreover, the function qn (z) is convex
where qn (z) =
0 t
univalent and is the best dominant of p ≺ qn in the sense that if p ≺ q, then
qn ≺ q.
An another generalization of Theorem A we refer to [5, p. 70]. However,
Theorem 2.5 cannot be written in the form presented in [5, p. 70] because h(z) =
{(1 + z)/(1 − z)}α is not convex. Therefore, Theorem 2.5 is a new extension
of Theorem A. For another generalization of Theorem A in this direction we
refer to [5, p. 70] and to [1], [10]. For φ(z) = 1/(βz + γ) Theorem 2.5 becomes
the following corollary, with the Briot-Bouquet differential subordination, for
related result we refer to [5, p. 81].
Corollary 2.6. Let h(z) = {(1 + z)(1 − z)}α , α ∈ (0, 1], and p(z) are analytic
in D with h(0) = p(0) = 1. Let φ(z) = 1/(βz + γ). Assume also that φ(p(z))
is analytic in D, moreover Re {φ(βh(z) + γ)} ≥ 0 in D. If
(12)
p(z) +
zp′ (z)
≺ h(z)
βp(z) + γ
(z ∈ D),
then
p(z) ≺ h(z)
For φ(z) = z
1/α
(z ∈ D).
Theorem 2.5 becomes the following corollary.
Corollary 2.7. Let h(z) = {(1 + z)/(1 − z)}α , α ∈ (0, 1], and p(z), p1/α (z)
are analytic in D with h(0) = p(0) = 1. If
(13)
p(z) + zp′ (z)p1/α (z) ≺ h(z)
(z ∈ D),
then
p(z) ≺ h(z)
(z ∈ D).
For α = 1/2 Corollary 2.7 becomes the following corollary.
p
Corollary 2.8. Let h(z) = (1 + z)/(1 − z) and p(z) are analytic in D with
h(0) = p(0) = 1. If
(14)
p(z) + zp′ (z)p2 (z) ≺ h(z)
(z ∈ D),
then
p(z) ≺ h(z)
(z ∈ D).
Theorem 2.9. Let p(z) be analytic in |z| < 1, with p(0) = −1 and suppose
that
zp′ (z)
1+z
(15)
− p(z) ≺
(z ∈ D).
p(z)
1−z
Then we have
1+z
(z ∈ D).
(16)
−p(z) ≺
1−z
A CRITERION FOR BOUNDED FUNCTIONS
223
Proof. If
1+z
for |z| < 1,
1−z
then by Lemma 1.1, there exist points
−p(z) 6≺
z0 = r0 eiθ ∈ D and ζ0 ∈ ∂D \ {−1} ,
for which
(17)
(18)
Re {−p(|z| < r0 )} > 0,
−p(z0 ) =
1 + ζ0
1 − ζ0
and
(19)
−z0 p′ (z0 ) = kζ0
2
(1 − ζ0 )2
for some k ≥ 1. By (18) and (19) we have
(20)
1 + ζ0
2kζ0
z0 p′ (z0 )
+
,
− p(z0 ) =
p(z0 )
1 − ζ02
1 − ζ0
furthermore,
z0 p′ (z0 )
1 + ζ0
2kζ0
Re
+
Re
− p(z0 ) = Re
p(z0 )
1 − ζ02
1 − ζ0
iIm {ζ0 }
ki
+ Re
= Re
Im {ζ0 }
1 − Re {ζ0 }
= 0.
This contradicts hypothesis (15) and it completes the proof.
Applying the above theorem, we have the following result.
Theorem 2.10. Let
∞
F (z) =
1 X
+
an z n
z n=1
be analytic in 0 < |z| < 1 and suppose that
zF ′ (z)
zF ′′ (z)
>0
−2
(21)
Re 1 + ′
F (z)
F (z)
(|z| < 1).
Then we have
(22)
zF ′ (z)
>0
Re −
F (z)
(z ∈ D).
224
MAMORU NUNOKAWA, SHIGEYOSHI OWA, AND JANUSZ SOKÓL
Proof. Let us put
p(z) =
zF ′ (z)
F (z)
p(0) = −1.
Then we have
p(z)F (z) = zF ′ (z),
hence
or
zF ′′ (z)
zp′ (z) zF ′ (z)
+
=1+ ′
,
p(z)
F (z)
F (z)
zp′ (z)
zF ′′ (z)
zF ′ (z)
− p(z) = 1 + ′
−2
.
p(z)
F (z)
F (z)
By (21) it has positive real part and hence by Theorem 2.9 we get (21).
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[9]
, New conditions for starlikeness and strongly starlikeness of order alpha, Houston Journal of Mathematics, in press.
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Mamoru Nunokawa
University of Gunma
Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan
E-mail address: mamoru− [email protected]
A CRITERION FOR BOUNDED FUNCTIONS
Shigeyoshi Owa
Kinki University
Higashi-Osaka, Osaka 577-8502, Japan
E-mail address: [email protected]
Janusz Sokól
Department of Mathematics
Rzeszów University of Technology
Al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland
E-mail address: [email protected]
225