Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 601597, 10 pages
doi:10.1155/2009/601597
Research Article
Conditions for Carathéodory Functions
Nak Eun Cho and In Hwa Kim
Department of Applied Mathematics, Pukyong National University, Busan 608-737, South Korea
Correspondence should be addressed to Nak Eun Cho, necho@pknu.ac.kr
Received 12 April 2009; Accepted 13 October 2009
Recommended by Yong Zhou
The purpose of the present paper is to derive some sufficient conditions for Carathéodory functions
in the open unit disk. Our results include several interesting corollaries as special cases.
Copyright q 2009 N. E. Cho and I. H. Kim. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let P be the class of functions p of the form
pz 1 ∞
pn zn ,
1.1
n1
which are analytic in the open unit disk U {z ∈ C : |z| < 1}. If p in P satisfies
Re pz > 0
z ∈ U,
1.2
then we say that p is the Catathéodory function.
Let A denote the class of all functions f analytic in the open unit disk U {z : |z| < 1}
with the usual normalization f0 f 0 − 1 0. If f and g are analytic in U, we say that f is
subordinate to g, written f ≺ g or fz ≺ gz, if g is univalent, f0 g0 and fU ⊂ gU.
For 0 < α ≤ 1, let STCα and STSα denote the classes of functions f ∈ A which are
strongly convex and starlike of order α; that is, which satisfy
zf z
1z α
≺
z ∈ U,
f z
1−z
zf z
1z α
≺
z ∈ U,
fz
1−z
1
1.3
1.4
2
Journal of Inequalities and Applications
respectively. We note that 1.3 and 1.4 can be expressed, equivalently, by the argument
functions. The classes STCα and STSα were introduced by Brannan and Kirwan 1 and
studied by Mocanu 2 and Nunokawa 3, 4. Also, we note that if α 1, then STSα
coincides with S∗ , the well-known class of starlikeunivalent functions with respect to
origin, and if 0 < α < 1, then STSα consists only of bounded starlike functions 1, and
hence the inclusion relation STSα ⊂ S∗ is proper. Furthermore, Nunokawa and Thomas
4 see also 5 found the value βα such that STCβα ⊂ STSα.
In the present paper, we consider general forms which cover the results by Mocanu
6 and Nunokawa and Thomas 4. An application of a certain integral operator is also
considered. Moreover, we give some sufficient conditions for univalent close-to-convex and
strongly starlike functions of order β as special cases of main results.
2. Main Results
To prove our results, we need the following lemma due to Nunokawa 3.
Lemma 2.1. Let p be analytic in U, p0 1 and pz /
0 in U. Suppose that there exists a point
z0 ∈ U such that
arg pz < π α
2
π
arg pz0 α
2
for |z| < |z0 |,
0 < α ≤ 1.
2.1
Then we have
z0 p z0 iαk,
pz0 2.2
1
π
1
k≥
x
when arg pz0 α,
2
x
2
1
π
1
k≤− x
when arg pz0 − α,
2
x
2
1/α
pz0 ±ix x > 0.
2.3
where
With the help of Lemma 2.1, we now derive the following theorem.
Theorem 2.2. Let p be nonzero analytic in U with p0 1 and let p satisfy the differential equation
ηzp z Bzpz a ibAz,
2.4
where η > 0, a ∈ R , 0 ≤ b ≤ a tanπ/2α, 0 < α < 1, Az signIm pz and Bz is analytic
in U with B0 a. If
arg Bz < π β η, α, a, b
2
z ∈ U,
2.5
Journal of Inequalities and Applications
3
where
2
SαT αa sinπ/2α − b cosπ/2α ηα
,
β η, α, a, b tan−1
π
SαT αa cosπ/2α b sinπ/2α
Sα 1 α1α/2 ,
T α 1 − α1−α/2 ,
2.6
2.7
then
arg pz < π α z ∈ U.
2
2.8
Proof. If there exists a point z0 ∈ U such that the conditions 2.1 are satisfied, then by
Lemma 2.1 we obtain 2.2 under the restrictions 2.3. Then we obtain
Az0 Bz0 ⎧
⎨1,
if pz0 ixα ,
⎩−1,
if pz0 −ixα ,
z0 p z0 a ibAz0 −η
pz0 pz0 a ibAz0 ±ix−α − iηαk
π b
a
π
α
cos
Az
sin
± α
0
xα
2
xα
2
b
a
π π
i α Az0 cos α − α sin ± α − ηαk .
x
2
x
2
2.9
Now we suppose that
1/α
ix
pz0 x > 0.
2.10
Then we have
arg Bz0 −tan−1
a sinπ/2α − b cosπ/2α ηαxα k
,
a cosπ/2α b sinπ/2α
2.11
where
kxα ≥
1 α1
x xα−1 ≡ gx x > 0.
2
2.12
4
Journal of Inequalities and Applications
Then,
by a simple calculation, we see that the function gx takes the minimum value at
x 1 − α/1 α. Hence, we have
arg Bz0 ≤ −tan
−1
1 α1α/2 1 − α1−α/2 a sinπ/2α − b cosπ/2α ηα
1 α1α/2 1 − α1−α/2 a cosπ/2α b sinπ/2α
2.13
π − β η, α, a, b ,
2
where βη, α, a, b is given by 2.6. This evidently contradicts the assumption of Theorem 2.2.
Next, we suppose that
pz0 1/α
−ix
2.14
x > 0.
Applying the same method as the above, we have
−1
arg Bz0 ≥ tan
1 α1α/2 1 − α1−α/2 a sinπ/2α − b cosπ/2α ηα
1 α1α/2 1 − α1−α/2 a cosπ/2α b sinπ/2α
π β η, α, a, b ,
2
2.15
where βη, α, a, b is given by 2.6, which is a contradiction to the assumption of Theorem 2.2.
Therefore, we complete the proof of Theorem 2.2.
Corollary 2.3. Let f ∈ A and η > 0, 0 < α < 1. If
π arg 1 − η zf z η 1 zf z
< β η, α z ∈ U,
2
fz
f z
2.16
where βη, α is given by 2.6 with a 1 and b 0, then f ∈ STSα.
Proof. Taking
fz
pz ,
zf z
zf z
zf z
η 1 Bz 1 − η
fz
f z
2.17
in Theorem 2.2, we can see that 2.4 is satisfied. Therefore, the result follows from
Theorem 2.2.
Corollary 2.4. Let f ∈ A and 0 < α < 1. Then STCβα ⊂ STSα, where βα is given by 2.6
with η a 1 and b 0.
Journal of Inequalities and Applications
5
By a similar method of the proof in Theorem 2.2, we have the following theorem.
Theorem 2.5. Let p be nonzero analytic in U with p0 1 and let p satisfy the differential equation
zp z
Bz a ibAz,
pz
2.18
where a ∈ R , b ∈ R− ∪ {0}, Az signIm pz, and Bz is analytic in U with B0 a. If
arg Bz < π αδ, a, b
2
z ∈ U,
2.19
where
αδ : αδ, a, b 2
δ−b
tan−1
π
a
δ > 0,
2.20
then
arg pz < π δ
2
z ∈ U.
2.21
Corollary 2.6. Let f ∈ STSαδ, where αδ is given by 2.20 with a 1 and b 0. Then
arg fz < π δ
z 2
z ∈ U.
2.22
Proof. Letting
pz z
,
fz
Bz zf z
fz
2.23
in Theorem 2.5, we have Corollary 2.6 immediately.
If we combine Corollaries 2.4 and 2.6, then we obtain the following result obtained by
Nunokawa and Thomas 4.
Corollary 2.7. Let f ∈ STCβδ, where
π
αδ
2
−1
tan αδ βδ tan
π
2
1 αδ1αδ/2 1 − αδ1−αδ/2 cosπ/2αδ
2.24
and αδ is given by 2.20. Then
arg fz < π δ
z 2
z ∈ U.
2.25
6
Journal of Inequalities and Applications
Corollary 2.8. Let f ∈ A, 0 < α < 1 and β, γ be real numbers with β /
0 and β γ > 0. If
arg β zf z γ < π δ α, β, γ
fz
2
z ∈ U,
2.26
where
α
2
π
−1
,
δ α, β, γ tan
tan α π
2
β γ 1 α1α/2 1 − α1−α/2 cosπ/2α
2.27
then
arg β zF z γ < π α z ∈ U,
2
Fz
2.28
where F is the integral operator defined by
Fz βγ
zγ
z
1/β
f β ttγ−1 dt
z ∈ U.
2.29
0
Proof. Let
Bz zf z
1
β
γ ,
βγ
fz
pz βγ
γ
z f β z
z
f β ttγ−1 dt.
2.30
2.31
0
Then Bz and pz are analytic in U with B0 p0 1. By a simple calculation, we have
1
zp z Bzpz 1.
βγ
2.32
Using a similar method of the proof in Theorem 2.2, we can obtain that
arg pz < π α z ∈ U.
2
2.33
From 2.29 and 2.31, we easily see that
1/β
Fz fz pz
.
2.34
Journal of Inequalities and Applications
7
Since
β
βγ
zF z
γ ,
Fz
pz
2.35
the conclusion of Corollary 2.8 immediately follows.
Remark 2.9. Letting α → 1 in Corollary 2.8, we have the result obtained by Miller and Mocanu
7.
The proof of the following theorem below is much akin to that of Theorem 2.2 and so
we omit for details involved.
Theorem 2.10. Let p be nonzero analytic in U with p0 1 and let p satisfy the differential equation
zp z
Bzpz a ibAz,
pz
2.36
where a ∈ R , b ∈ R− ∪ {0}, Az signIm pz and Bz is analytic in U with B0 a. If
arg Bz < π βα, a, b
2
z ∈ U,
2.37
0 < α ≤ 1,
2.38
where
βα, a, b α α−b
2
tan−1
π
a
then
arg pz < π α z ∈ U.
2
2.39
Corollary 2.11. Let f ∈ A with f z /
0 in U and 0 < α ≤ 1. If
arg f z zf z < π βα
2
z ∈ U,
2.40
where βα is given by 2.38 with a 1 and b 0, then
arg f z < π α z ∈ U,
2
that is, f is univalent (close-to-convex) in U.
2.41
8
Journal of Inequalities and Applications
Proof. Let
pz 1
,
f z
Bz f z zf z
2.42
in Theorem 2.10. Then 2.36 is satisfied and so the result follows.
By applying Theorem 2.10, we have the following result obtained by Mocanu 6.
Corollary 2.12. Let f ∈ A with fz/z /
0 and α0 be the solution of the equation given by
2α 2
tan−1 α 1 0 < α < 1.
π
2.43
If
arg f z < π 1 − α0 z ∈ U,
2
2.44
then f ∈ S∗ .
Proof. Let
pz z
,
fz
Bz f z.
2.45
Then, by Theorem 2.10, condition 2.44 implies that
arg z < π α0 .
fz 2
2.46
arg zf z ≤ arg f z arg z < π ,
fz
fz 2
2.47
Therefore, we have
which completes the proof of Corollary 2.12.
0 in U and 0 < α ≤ 1. If
Corollary 2.13. Let f ∈ A with fzf z/z /
arg zf z 2 zf z − zf z < π βα
fz
f z
fz 2
where βα is given by 2.38, then f ∈ STSα.
z ∈ U,
2.48
Journal of Inequalities and Applications
9
Finally, we have the following result.
Theorem 2.14. Let p be nonzero analytic in U with p0 1. If
arg 1 − λpz λzp z < π βλ, α,
2
βλ, α α λα
2
tan−1
π
1−λ
0 ≤ λ < 1; 0 < α < 1,
2.49
2.50
then
arg pz < π α z ∈ U.
2
2.51
Proof. If there exists a point z0 ∈ U satisfying the conditions of Lemma 2.1, then we have
1 − λpz0 λz0 p z0 ±ixα 1 − λ iλαk.
2.52
Now we suppose that
1/α
ix
pz0 x > 0.
2.53
Then we have
π
λαk
arg 1 − λpz0 λz0 p z0 α tan−1
2
1−λ
λα
π
2
≥
α tan−1
2
π
1−λ
2.54
π
βλ, α,
2
where βλ, α is given by 2.50. Also, for the case
pz0 1/α
−ix
x > 0,
2.55
we obtain
λα
π
2
arg 1 − λpz0 λz0 p z0 ≤ −
α tan−1
2
π
1−λ
π
− βλ, α,
2
2.56
where βλ, α is given by 2.50. These contradict the assumption of Theorem 2.14 and so we
complete the proof of Theorem 2.14.
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Journal of Inequalities and Applications
0 in U and 0 < α < 1. If
Corollary 2.15. Let f ∈ A with fzf z/z /
π
< α 1
arg zf z 1 zf z − zf z
2
fz
f z
fz
z ∈ U,
2.57
then f ∈ STSα.
Acknowledgment
This research was supported by Basic Science Research Program through the National
Research Foundation of Korea NRF funded by the Ministry of Education, Science and
Technology No. 2009-0066192.
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