Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 391038, 8 pages
doi:10.1155/2012/391038
Research Article
Argument Property for Certain Analytic Functions
Qing Yang and Jin-Lin Liu
Department of Mathematics, Yangzhou University, Yangzhou 225002, China
Correspondence should be addressed to Jin-Lin Liu, jlliu@yzu.edu.cn
Received 18 September 2011; Accepted 1 November 2011
Academic Editor: Khalida Inayat Noor
Copyright q 2012 Q. Yang and J.-L. Liu. 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.
n
Let P be the class of functions pz of the form pz 1 ∞
n1 cn z which are analytic in the open
unit disk U {z : |z| < 1}. The object of the present paper is to derive certain argument inequalities
of analytic functions pz in P.
1. Introduction
Let P be the class of functions pz of the form
pz 1 ∞
cn zn ,
1.1
n1
which are analytic in the open unit disk U {z : |z| < 1}. For functions p and g in the class
P , we say that p is subordinate to g if there exists an analytic function w in U with w0 0,
|wz| < 1 z ∈ U, and such that pz gwz z ∈ U. We denote this subordination by
p≺g
z ∈ U.
1.2
If g is univalent in U, then this subordination p ≺ g is equivalent to p0 g0 and pU ⊂
gU.
Recently, several authors investigated various argument properties of analytic functions see, e.g., 1–6. The object of the present paper is to discuss some argument inequalities
for p in the class P .
2
Abstract and Applied Analysis
Throughout this paper, we let
0 < α1 ≤ 1,
0 < α2 ≤ 1,
β
α1 − α2
,
α1 α2
c eβπi .
1.3
In order to prove our main result, we well need the following lemma.
Lemma 1.1 see 6. Let λ0 , λ, a, b ∈ R and μ ∈ C. Also let
λ0 a ≥ 0,
b 1 Re μ ≥ 0,
λb 2 ≥ 0,
1.4
1
.
|a − b − 1| ≤
max{α1 , α2 }
2
,
|b 1| ≤
α1 α2
If q ∈ P satisfies
b2
b1
b
a
μ qz
zq z qz ≺ hz
λ0 qz λ qz
z ∈ U,
1.5
where
1 cz
1−z
hz λ0
×
aα1 α2 /2
1 cz
μλ
1−z
1 cz
1−z
α1 α2 /2
1/2b1α1 α2 α1 α2
2
cz
z
1 − z 1 cz
1.6
is (close to convex) univalent, then
−
α1 π
α2 π
< arg qz <
2
2
z ∈ U.
1.7
The bounds α1 and α2 in 1.7 are sharp for the function q defined by
qz 1 cz
1−z
α1 α2 /2
.
1.8
Remark 1.2 see 6. The function q defined by 1.8 is analytic and univalently convex in U
and
α1 π α2 π
< arg w <
.
qU w : w ∈ C, −
2
2
1.9
Abstract and Applied Analysis
3
2. Main Result
Our main theorem is given by the following.
Theorem 2.1. Let
λ0 > 0,
1
,
max{α1 , α2 }
0<a≤
|b 1| ≤
2
,
α1 α2
0≤a−b−1≤
1
.
max{α1 , α2 }
2.1
If p ∈ P satisfies
b γ1 π
a
γ2 π
< arg λ0 pz zp z pz
<
2
2
−
z ∈ U,
2.2
where
γj γj a, b, α1 , α2 α1 α2 /2 cos βπ/2 cos a − b − 1αj π/2
2
−1
aαj tan
π
2λ0 δj a, b, α1 , α2 α1 α2 /2 cos βπ/2 sin a − b − 1αj π/2
j 1, 2 ,
δj a, b, α1 , α2 1A
2 1/2
1 − A cos βπ/2
−1j sin βπ/2
2 1/2
2
A
j
2 1 − A cos βπ/2
−1 1 − A cos βπ/2
sin βπ/2
j 1, 2 ,
2.3
where A denotes a − b − 1α1 α2 /2, then
−
α1 π
α2 π
< arg pz <
2
2
z ∈ U.
2.4
The bounds γ1 and γ2 in 2.2 are the largest numbers such that 2.4 holds true.
Proof. By taking λ μ 0 in Lemma 1.1, we find that if p ∈ P satisfies
b
a
λ0 pz zp z pz ≺ hz
z ∈ U,
2.5
where
hz 1 cz
1−z
b1α1 α2 /2 cz
1 cz a−b−1α1 α2 /2 α1 α2
z
λ0
, 2.6
1−z
2
1 − z 1 cz
then 2.4 holds true.
4
Abstract and Applied Analysis
1 and z /
− 1/c, we get
For z eiθ , z /
θ βπ
cz
1 i
tan
,
1 cz 2 2
2
cos θ βπ /2 α1 πi/α1 α2 e
0.
/
sinθ/2
1 i
θ
z
− cot ,
1−z
2 2
2
1 cz 1 eiθβπ
1−z
1 − eiθ
2.7
We consider the following two cases.
i If
kθ cos
θ βπ
θ
sin > 0,
2
2
2.8
then from 2.7, and 2.6, we have
h eiθ b1α1 α2 /2
cos θ βπ /2
· eb1α1 πi/2
sinθ/2
⎛ a−b−1α1 α2 /2
⎞
α
cos θ βπ /2
cos
βπ/2
α
1
2
⎠,
× ⎝ λ0
· ea−b−1α1 πi/2 i
sinθ/2
4kθ
2.9
and so
1
arg h eiθ
aα1 π tan−1
2
α1 α2 /2 cos βπ/2 cosa − b − 1α1 π/2
,
2λ0 k1 θ α1 α2 /2 cos βπ/2 sina − b − 1α1 π/2
2.10
where λ0 > 0, 0 ≤ a − b − 1α1 α2 ≤ 2, eiθ /
1, eiθ /
− 1/c,
k1 θ kθ
a−b−1α1 α2 /2
cos θ βπ /2
> 0.
sinθ/2
2.11
We now calculate the maximum value of k1 θ. It is easy to verify that
lim k1 θ θ→0
lim k1 θ 0
eiθ → −1/c
2.12
Abstract and Applied Analysis
5
and that
k1 θ
a−b−1α1 α2 /2−1
cos βπ/2
cos θ βπ /2
·
kθ
sinθ/2
sinθ/22
a−b−1α1 α2 /2
βπ
1 cos θ βπ /2
· cos θ 2
sinθ/2
2
a − b − 1α1 α2 −
4
1
2
a−b−1α1 α2 /2
cos θ βπ /2
.
sinθ/2
2.13
βπ
βπ
a − b − 1α1 α2 cos
× cos θ −
.
2
2
2
Set
θ1 cos−1
βπ
a − b − 1α1 α2 cos
2
2
−
βπ
,
2
2.14
then k1 θ1 0. Noting that
0 ≤ a − b − 1α1 α2 ≤ 2, −1 < β < 1,
βπ
βπ
π
a − b − 1α1 α2 < cos−1
cos
< ,
2
2
2
2
2.15
θ1 βπ π
< .
2
2
2.16
we easily have
0 < θ1 < π,
0 < θ1 βπ π
< ,
2
2
0<
Hence, kθ1 > 0, and it follows from 2.11 to 2.16 that
0 < k1 θ ≤ k1 θ1 −a−b−1α1 α2 1A
θ1 βπ
θ1
sin
· cos
2
2
−A 1A
βπ
βπ
1 − cos θ1
1
·
sin θ1 − sin
2
2
2
2
1A
2 1/2
− sin βπ/2
1 − A cos βπ/2
2 2 1/2
A
2 1 − A cos βπ/2
− 1 − A cos βπ/2
sin βπ/2
sin
θ1
2
δ1 a, b, α1 , α2 ,
2.17
6
Abstract and Applied Analysis
where A denotes a − b − 1α1 α2 /2. Thus, by using 2.1, 2.10, and 2.17, we arrive at
≥ arg h eiθ1
π > arg h eiθ
1
aα1 π tan−1
2
α1 α2 /2 cos βπ/2 cosa − b − 1α1 π/2
2λ0 δ1 a, b, α1 , α2 α1 α2 /2 cos βπ/2 sina − b − 1α1 π/2
γ1 π
> 0.
2
2.18
ii If kθ < 0, then we obtain
h eiθ b1α1 α2 /2
cos θ βπ /2
−
· e−b1α2 πi/2
sinθ/2
⎛ a−b−1α1 α2 /2
⎞
cos θ βπ /2
α
cos
βπ/2
α
1
2
⎠,
× ⎝λ0 −
· e−a−b−1α2 πi/2 i
sinθ/2
4kθ
2.19
which leads to
1
arg h eiθ − aα2 π − tan−1
2
α1 α2 /2 cos βπ/2 cosa − b − 1α2 π/2
,
2λ0 k2 θ α1 α2 /2 cos βπ/2 cosa − b − 1α2 π/2
2.20
where λ0 > 0, 0 ≤ a − b − 1α1 α2 ≤ 2, eiθ /
1, eiθ /
− 1/c,
a−b−1α1 α2 /2
cos θ βπ /2
k2 θ −kθ −
> 0.
sinθ/2
2.21
Now, we have
lim k2 θ θ→0
k2 θ 1
2
lim k2 θ 0,
eiθ → −1/c
a−b−1α1 α2 /2
cos θ βπ /2
βπ
βπ
a − b − 1α1 α2 −
cos
− cos −θ −
.
sinθ/2
2
2
2
2.22
Let
θ2 −cos
−1
βπ
a − b − 1α1 α2 cos
2
2
−
βπ
,
2
2.23
Abstract and Applied Analysis
7
then k2 θ2 0, θ1 θ2 −βπ,
−π < θ2 < 0,
−
βπ
π
< θ2 < 0,
2
2
−
π θ2 βπ
<
< 0.
2
2
2.24
Hence, we deduce that kθ2 < 0 and
0 < k2 θ ≤ k2 θ2 −a−b−1α1 α2 1A
θ2 βπ
θ2
sin
· − cos
2
2
−A 1A
βπ
βπ
1 − cos θ2
1
sin
− sin θ2 ·
2
2
2
2
1A
2 1/2
sin βπ/2
1 − A cos βπ/2
2 2 1/2
A
2 1 − A cos βπ/2
1 − A cos βπ/2
sin βπ/2
− sin
θ2
2
2.25
δ2 a, b, α1 , α2 ,
where A a − b − 1α1 α2 /2. Further, by using 2.1, 2.20, and 2.25, we find that
−π < arg h eiθ
≤ arg h eiθ2
1
− aα2 π − tan−1
2
−
α1 α2 /2 cos βπ/2 cosa − b − 1α2 π/2
2λ0 δ2 a, b, α1 , α2 α1 α2 /2 cos βπ/2 cosa − b − 1α2 π/2
γ2 π
< 0.
2
2.26
In view of h0 1 > 0, we conclude from 2.18 and 2.26 that hU properly contains the
angular region −γ2 π/2 < arg w < γ1 π/2 in the complex w-plane. Therefore, if p ∈ P satisfies
2.2, then the subordination relation 2.5 holds true, and thus we arrive at 2.4.
Furthermore, for the function q defined by 1.8, we have
α1 π
α2 π
< arg qz <
z ∈ U,
2
2
b
a
λ0 qz zq z qz hz.
−
2.27
Hence, by using 2.18 and 2.25, we see that the bounds γ1 and γ2 in 2.2 are best possible.
8
Abstract and Applied Analysis
Acknowledgment
The authors would like to express sincere thanks to the referees for careful reading and suggestions which helped them to improve the paper.
References
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Academy. Series A, vol. 69, no. 7, pp. 234–237, 1993.
3 M. Nunokawa, S. Owa, E. Y. Duman, and M. Aydoğan, “Some properties of analytic functions relating
to the Miller and Mocanu result,” Computers & Mathematics with Applications, vol. 61, no. 5, pp. 1291–
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4 N. Takahashi and M. Nunokawa, “A certain connection between starlike and convex functions,” Applied Mathematics Letters, vol. 16, no. 5, pp. 653–655, 2003.
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