Journal of Inequalities in Pure and
Applied Mathematics
APPLICATIONS OF NUNOKAWA’S THEOREM
A.Y. LASHIN
Department of Mathematics
Faculty of Science Mansoura University,
Mansoura 35516, EGYPT.
volume 5, issue 4, article 111,
2004.
Received 29 August, 2004;
accepted 13 December, 2004.
Communicated by: H.M. Srivastava
EMail: aylashin@yahoo.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
The object of the present paper is to give applications of the Nunokawa Theorem [Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 234-237]. Our results
have some interesting examples as special cases .
2000 Mathematics Subject Classification: 30C45
Key words: Analytic functions, Univalent functions, Subordination
Applications of Nunokawa’s
Theorem
The author thank the referee for his helpful suggestion.
A.Y. Lashin
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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3
5
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1.
Introduction
Let A be the class of functions of the form
(1.1)
f (z) = z +
∞
X
an z n
n=2
which are analytic in the open unit disk U = {z : |z| < 1}. It is known that the
class
)
(
(
µ−1 )
f
(z)
> 0, µ > 0, z ∈ U
(1.2) B(µ) = f (z) ∈ A : Re f 0 (z)
z
is the class of univalent functions in U ([3]).
To derive our main theorem, we need the following lemma due to Nunokawa
[2].
Lemma 1.1. Let p(z) be analytic in U , with p(0) = 1 and p(z) 6= 0 (z ∈ U ). If
there exists a point z0 ∈ U, such that
π
|arg p(z)| < α for |z| < |z0 |
2
and
|arg p(z0 )| =
then we have
π
α (α > 0),
2
z0 p0 (z0 )
= ikα,
p(z0 )
where k ≥ 1 when arg p(z0 ) = π2 α and k ≤ −1 when arg p(z0 ) = − π2 α
Applications of Nunokawa’s
Theorem
A.Y. Lashin
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In [1] , Miller and Mocanu proved the following theorem.
Theorem A. Let β0 = 1.21872..., be the solution of
3
βπ = π − tan−1 β
2
and let
α = α(β) = β +
2
tan−1 β
π
Applications of Nunokawa’s
Theorem
for 0 < β < β0 .
If p(z) is analytic in U , with p(0) = 1, then
0
p(z) + zp (z) ≺
or
1+z
1−z
α
A.Y. Lashin
⇒ p(z) ≺
1+z
1−z
β
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π
π
|arg (p(z) + zp0 (z))| < α ⇒ |arg p(z)| < β.
2
2
Corresponding to Theorem A, we will obtain a result which is useful in
obtaining applications of analytic function theory.
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2.
Main Results
Now we derive:
Theorem 2.1. Let p(z) be analytic in U , with p(0) = 1 and p(z) 6= 0 (z ∈ U )
and suppose that
π
2
0
−1
|arg (p(z) + βzp (z))| <
α + tan βα
(α > 0, β > 0),
2
π
then we have
π
α for z ∈ U.
2
Proof. If there exists a point z0 ∈ U, such that
|arg p(z)| <
|arg p(z)| <
π
α for |z| < |z0 |
2
and
π
|arg p(z0 )| = α (α > 0),
2
then from Lemma 1.1, we have
(i) for the case arg p(z0 ) = π2 α,
z0 p0 (z0 )
0
arg (p(z) + βz0 p (z0 )) = arg p(z0 ) 1 + β
p(z0 )
π
π
= α + arg (1 + iβαk) ≥ α + tan−1 βα.
2
2
Applications of Nunokawa’s
Theorem
A.Y. Lashin
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J. Ineq. Pure and Appl. Math. 5(4) Art. 111, 2004
This contradicts our condition in the theorem.
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(ii) for the case arg p(z0 ) = − π2 α, the application of the same method as in (i)
shows that
π
0
−1
arg (p(z) + βz0 p (z0 )) ≤ −
α + tan βα .
2
This also contradicts the assumption of the theorem, hence the theorem is proved.
Making p(z) = f 0 (z) for f (z) ∈ A in Theorem 2.1, we have
Applications of Nunokawa’s
Theorem
Example 2.1. If f (z) ∈ A satisfies
π
|arg (f (z) + βzf (z))| <
2
0
00
2
α + tan−1 βα
π
then we have
π
|arg f (z)| < α ,
2
where α > 0, β > 0 and z ∈ U .
0
Further, taking p(z) =
f (z)
z
for f (z) ∈ A in Theorem 2.1, we have
Example 2.2. If f (z) ∈ A satisfies
π
f
(z)
2
0
−1
arg{(1 − β)
+ βf (z)} <
α + tan βα ,
z
2
π
then we have
π
f
(z)
< α,
arg
z 2
where α > 0, 0 < β ≤ 1 and z ∈ U .
A.Y. Lashin
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Theorem 2.2. If f (z) ∈ A satisfies
µ−1 f
(z)
2
π
0
−1 α
α + tan
,
<
arg f (z)
2
z
π
µ
then we have
µ arg f (z) < π α,
2
z
where α > 0, µ > 0 and z ∈ U .
n
oµ
Proof. Let p(z) = f (z)
, µ > 0, then we have
z
1
p(z) + zp0 (z) = f 0 (z)
µ
f (z)
z
Applications of Nunokawa’s
Theorem
A.Y. Lashin
µ−1
and the statements of the theorem directly follow from Theorem 2.1.
Theorem 2.3. Let µ > 0 , c + µ > 0 and α > 0. If f (z) ∈ A satisfies
µ−1 f
(z)
2
α
π
0
−1
α + tan
, (z ∈ U )
arg f (z)
<
2
z
π
µ+c
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then F (z) = [Iµ,c (f )](z) defined by
Iµ,c f (z) =
µ+c
zc
Z
0
µ1
z
f µ (t)tc−1 dt ,
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([Iµ,c (f )](z)/z 6= 0 in U )
J. Ineq. Pure and Appl. Math. 5(4) Art. 111, 2004
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satisfies
µ−1 F
(z)
π
arg F 0 (z)
< α.
2
z
Proof. Consider the function p defined by
0
p(z) = F (z)
F (z)
z
µ−1
(z ∈ U ).
Applications of Nunokawa’s
Theorem
Then we easily see that
1
p(z) +
zp0 (z) = f 0 (z)
µ+c
f (z)
z
A.Y. Lashin
µ−1
,
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and the statements of the theorem directly follow from Theorem 2.1.
Theorem 2.4. Let a function f (z) ∈ A satisfy the following inequalities
µ+1 z
2
π
0
−1 α
(2.1)
−α + tan
, (z ∈ U )
arg f (z)
<
2
f (z)
π
µ
for some α (0 < α ≤ 1), (0 < µ < 1). Then
µ π
f
(z)
arg
< α.
2
z
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µ
Proof. Let us define the function p(z) by p(z) = f (z)
, (0 < µ < 1). Then
z
p(z) satisfies
µ+1
z
1
1 zp0 (z)
0
f (z)
=
1+
.
f (z)
p(z)
µ p(z)
If there exists a point z0 ∈ U, such that
π
|arg p(z)| < α for |z| < |z0 |
2
and
π
|arg p(z0 )| = α,
2
then from Lemma 1.1, we have:
(i) for the case arg p(z0 ) = π2 α,
µ+1
1
1 zp0 (z0 )
z
0
arg f (z0 )
= arg
1+
f (z0 )
p(z0 )
µ p(z0 )
π
iαk
= − α + arg 1 +
2
µ
π
α
≥ − α + tan−1 .
2
µ
This contradicts our condition in the theorem.
(ii) for the case arg p(z0 ) = − π2 α, the application of the same method as in (i)
shows that
µ+1
π
z
0
−1 α
≤ − − α + tan
.
arg f (z0 )
f (z0 )
2
µ
Applications of Nunokawa’s
Theorem
A.Y. Lashin
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This also contradicts the assumption of the theorem, hence the theorem is proved.
Theorem 2.5. Let f (z) ∈ A satisfy the condition (2.1) and let
(2.2)
c−µ
F (z) =
z c−µ
Z
z
0
t
f (t)
µ
− µ1
dt
,
where c − µ > 0. Then
Applications of Nunokawa’s
Theorem
µ+1 z
π
< α.
arg F 0 (z)
2
F (z)
A.Y. Lashin
Proof. If we put
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p(z) = F 0 (z)
z
F (z)
µ+1
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then from (2.2) we have
1
p(z) +
zp0 (z) = f 0 (z)
c−µ
z
f (z)
µ+1
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The statements of the theorem then directly follow from Theorem 2.1.
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References
[1] S.S. MILLER AND P.T. MOCANU, Differential Subordinations, Marcel
Dekker, INC., New York, Basel, 2000.
[2] M. NUNOKAWA, On the order of strongly convex functions, Proc. Japan
Acad. Ser. A Math. Sci., 69 (1993), 234–237.
[3] M. OBRADOVIĆ, A class of univalent functions, Hokkaido Math. J., 27
(1988), 329–335.
Applications of Nunokawa’s
Theorem
A.Y. Lashin
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