Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 679704, 12 pages
doi:10.1155/2011/679704
Research Article
Certain Conditions for Starlikeness of Analytic
Functions of Koebe Type
Saibah Siregar1 and Maslina Darus2
1
Faculty of Science and Biotechnology, Universiti Selangor, Bestari Jaya,
45600 Selangor Darul Ehsan, Malaysia
2
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
Bangi, 43600 Selangor Darul Ehsan, Malaysia
Correspondence should be addressed to Maslina Darus, maslina@ukm.my
Received 1 May 2011; Revised 29 June 2011; Accepted 5 July 2011
Academic Editor: A. Zayed
Copyright q 2011 S. Siregar and M. Darus. 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.
For α ≥ 0, λ > 0, we consider the Mα, λb of normalized analytic α − λ convex functions defined in
the open unit disc U. In this paper, we investigate the class Mα, λb , that is, Re{zfb z/fb z1 −
α α1 − λzfb z/fb z αλ1 zfb z/fb z} > 0, with fb is Koebe type, that is, fb z :
z/1−zn b . The subordination result for the aforementioned class will be given. Further, by making
use of Jack’s Lemma as well as several differential and other inequalities, the authors derived
sufficient conditions for starlikeness of the class Mα, λb of n-fold symmetric analytic functions
of Koebe type. Relevant connections of the results presented here with those given in the earlier
works are also indicated.
1. Introduction
Let A denote the class of normalized analytic functions of the form
∞
fz z ak zk ,
1.1
k2
which are analytic in the open unit disk U {z : |z| < 1}. Also, as usual, let
zf z
S∗ f : f ∈ A, Re
> 0, z ∈ U ,
fz
zf z
> 0, z ∈ U ,
K f : f ∈ A, Re 1 f z
be the familiar classes of starlike functions in U and convex functions in U, respectively.
1.2
2
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If the functions f and g are analytic in U, then we say that the function f is subordinate
to g, or g is superordinate to f written as f ≺ g if there exist a function wz analytic U, such
that |wz| < 1 and z ∈ U, and w0 0 with fz gwz in U. If g is univalent in U, then
f ≺ g is equivalent to f0 g0 and fU ⊆ gU.
Next, we let the Mα, that is,
Mα zf z
zf z
α 1 > 0, z ∈ U .
fz ∈ A : Re 1 − α
fz
f z
1.3
The class Mα was first introduced by Mocanu 1, which was then known as the class of
α convex or α-starlike functions. Later, Miller et al. 2 studied this class and showed that
Mα is a subclass of S∗ for any real number α and also that Mα is a subclass of K for
α ≥ 1. We note that M0 S∗ and M1 K. Note also that Mocanu introduced Mα with
fz · f z/0. But Sakaguchi and Fukui 3 later showed that this condition was not needed.
Motivated essentially by the aforementioned earlier works, we aim here at deriving
sufficient conditions for starlikeness of n-fold symmetric function fb of the Koebe type,
defined by
fb z :
z
1 − zn b
b ≥ 0; n ∈ N : {1, 2, 3, . . .},
1.4
which obviously corresponds to the familiar Koebe function when n 1 and b 2.
Definition 1.1. A function fz given by 1.1 is said to be in the class Mα, λb , for α ≥ 0,
λ > 0, if the following conditions are satisfied:
zfb z
zfb z
zfb z
1 − α α1 − λ
αλ 1 > 0,
Re
fb z
fb z
fb z
z ∈ U.
1.5
In this paper, we consider the class of functions Mα, λb .
In addition, in this paper, authors investigate the subordination of the class denoted
by Mα, λb .
We have the following inclusion relationships:
i M0, λ1 ⊂ S∗
ii Mα, 11 ⊂ Hα ⊂ S∗ , which Hα has studied by 4.
The work of Siregar et al. 5 and Bansal and Raina 6 have also motivated us to come to
these problems. Look also at 7, 8 for different studies.
The following result popularly known as Jack’s Lemma will also be required in the
derivation of our result Theorem 4.1 below.
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3
2. Preliminaries
Lemma 2.1 see 9. Let qz be univalent in U and let the function θ and φ be analytic in a domain
D containing qU, with φw /
0 when w ∈ qU. Set
Qz γzq zφ qz ,
γ > 0,
2.1
hz θ qz Qz,
and suppose that
i Qz is univalent and starlike in U;
ii Rezh z/Qz Reθ qz/φqz zQ z/Qz > 0, z ∈ U.
If pz is analytic in U with p0 q0 1, pU ⊂ D, and
θ pz γzp zφ pz ≺ θ qz γzq zφ qz hz,
2.2
pz ≺ qz,
2.3
then
and q is the best dominant.
Lemma 2.2 see 10. Let the (nonconstant) function wz be analytic in U such that w0 0. If
|wz| attains its maximum value on circle |z| r < 1 at a point zo ∈ U, we have
zo w zo kwzo ,
2.4
where k ≥ 1 is a real number.
3. The Subordination Result
Theorem 3.1. Let fz ∈ A satisfy fz /
0 z ∈ U. Also, let the function qz be univalent in U,
with q0 1 and qz 0,
for
λ
>
0
and
α
≥ 0, such that
/
zq z
>0
Re 1 q z
z ∈ U,
zq z
Re λ 2qz >0
q z
3.1
z ∈ U.
If
zfb z
zfb z
zfb z
1 − α α1 − λ
αλ 1 ≺ hz
fb z
fb z
fb z
z ∈ U,
3.2
4
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where
2
hz α qz 1 − α qz αλzq z,
3.3
then
zfb z
≺ qz
fb z
z ∈ U,
3.4
and qz is the best dominant of 3.2.
Proof. We first choose
pz zfb z
,
fb z
θw w1 − α αw,
φw 1,
3.5
then θw and φw are analytic inside the domain D∗ , which contains qU, q0 1, and
φw /
0 when w ∈ qU.
Now, if we define the functions Qz and hz by
Qz αλzq zφ qz αλzq z,
2
hz θ qz Qz α qz 1 − α qz αλzq z,
3.6
then it follows from 3.1 that Qz is starlike in U and
zh z
> 0 z ∈ U.
Re
Qz
3.7
We also note that the function pz is analytic in U, with p0 q0 1. Since 0 ∈
/ pU,
∗
therefore, pU ⊂ D , γ αλ > 0 and hence, the hypothesis of Lemma 2.1 are satisfied.
Applying Lemma 2.1, we find that
zfb z
zfb z
zfb z
1 − α α1 − λ
αλ 1 fb z
fb z
fb z
2
α pz 1 − α pz αλzp z θ pz αλzp z ≺ hz
2
α qz 1 − α qz αλzq z θ qz αλzq z,
3.8
z ∈ U,
which implies that
zfb z
≺ qz
fb z
and qz is the best dominant of 3.2.
z ∈ U,
3.9
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5
4. The Properties of the Class Mα, λb
We begin by proving a stronger result than what we indicated in the preceding section.
Theorem 4.1. Let the n-fold symmetric function fb z, defined by 1.4, be analytic in U, with
fb z
0
/
z
z ∈ U.
4.1
If fb z satisfies the inequality:
zfb z
zfb z
zfb z
αnb
αλnb
nb
Re
1−
−
,
> 1−
1 − α α1 − λ
αλ 1 fb z
fb z
2
2
4
fb z
4.2
z ∈ U, then fb z is starlike in U for
α > 0, λ > 0,
√ √
αλ 2α 2 Δ
αλ 2α 2 − Δ
≤ nb ≤
,
2α
2α
Δ : α2 λ 22 4αλ − 2 4 .
4.3
If fb z satisfies the inequality 4.2 with λ 1, that is, if
Re
αz2 fb z zf z
fb z
fz
>−
αnb
nb
αnb
1−
1−
,
4
2
2
z ∈ U
4.4
then fb z is starlike in U for
α > 0,
3α 2 −
2α
√
Δ∗
≤ nb ≤
√
3α 2 Δ∗
,
2α
Δ∗ : 9α2 − 4α 4 .
4.5
Proof. Let α > 0, λ > 0 and fb z satisfy the hypothesis of Theorem 4.1. We put
zfb z 1 nb − 1wz
,
fb z
1 − wz
4.6
where wz is analytic in U, with
w0 0,
wz /
1,
z ∈ U,
4.7
such that, we can write
1
zfb z
fb z
nbzw z
1 nb − 1wz
,
1 − wz
1 − wz1 nb − 1wz
4.8
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International Journal of Mathematics and Mathematical Sciences
which, in turn, implies that
zfb z
zfb z
zfb z
1 − α α1 − λ
αλ 1 fb z
fb z
fb z
1 nb − 1wz
1 − wz
1 nb − 1wz
× 1 − α α1 − λ
1 − wz
nbzw z
1 nb − 1wz
αλ
1 − wz
1 − wz1 nb − 1wz
1 nb − 1wz
1 nb − 1wz2 λnbzw z
α
1 − α
.
1 − wz
1 − wz2
4.9
Now, we claim that |wz| < 1 z ∈ U. If there exists a zo in U such that |wzo | 1, then by
Jack’s Lemma Lemma 2.2, we have
zo w zo kwzo ,
4.10
where k ≥ 1 is a real number.
By setting wzo eiθ 0 ≤ θ ≤ 2π, thus, we find that
zo fb zo zo fb zo zo fb zo Re
1 − α α1 − λ
αλ 1 fb zo fb zo fb zo 1 nb − 1wzo Re 1 − α
1 − wzo ⎧
⎨
Re 1 − α
⎩
1 nb − 1eiθ
1 − eiθ
α
λnbzo w zo 1 nb − 1wzo 2
1 − wzo 2
2 ⎞⎫
⎬
λnbkeiθ 1 nb − 1eiθ
⎠
α⎝
2
⎭
1 − eiθ
⎛
1 αλnbk αnb nb − 2eiθ nb − 1αnb − 1 − 1 − αei2θ
Re
2
1 − eiθ
2αnbλk 2−nb4nb−8 cos θ nb−1αnb−11 cos 2θ−αnbλk3−2nb−3nb6
23 − 4 cos θ 2 cos 2θ
≤
nb
1−
2
since k ≥ 1.
αnb
1−
2
−
αλnb
,
4
z ∈ U,
4.11
International Journal of Mathematics and Mathematical Sciences
7
If we let
zo fb zo zo fb zo zo fb zo Re
αλ 1 1 − α α1 − λ
fb zo fb zo fb zo ≤
αnb
αλnb
nb
1−
−
1−
2
2
4
4.12
τnb,
then
τnb ≤ 0,
√
√
αλ 2α 2 Δ
αλ 2α 2 Δ; Δ
2
2
≤ nb ≤
: α λ 2 4αλ − 2 4 .
2α
2α
4.13
Thus, we have
zfb z
zfb z
zfb z
Re
≤0
1 − α α1 − λ
αλ 1 fb z
fb z
fb z
z ∈ U,
√
√
αλ 2α 2 Δ
αλ 2α 2 Δ
2
2
≤ nb ≤
; Δ : α λ 2 4αλ − 2 4 ,
2α
2α
4.14
4.15
which is a contradiction to the hypotheses of 4.2.
Therefore, |wz| < 1 for all z in U. Hence fb is starlike in U, then by proving the
assertion i of Theorem 4.1, this completes the proof of our theorem.
Next, we arrive to the following remark which was given by Fukui et al. 11, and so
we omit the detail here.
Remark 4.2. Let the n-fold symmetric function fb z, defined by 1.4, be analytic in U, with
fb z
0
/
z
z ∈ U.
4.16
If fb z satisfies the inequality 4.2 with α 0, that is, if
Re
zfb z
fb z
>1−
nb
2
z ∈ U,
then fb z is starlike in U for 0 ≤ nb < 2.
The following remark was obtained by Kamali and Srivastava 12.
4.17
8
International Journal of Mathematics and Mathematical Sciences
Remark 4.3. Let the n-fold symmetric function fb z, defined by 1.4, be analytic in U, with
fb z
0
/
z
z ∈ U.
4.18
If fb z satisfies the inequality 4.2 with λ 1, that is, if
Re
αz2 fb z zf z
fb z
fz
αnb
nb
αnb
>−
1−
1−
4
2
2
z ∈ U,
4.19
then fb z is starlike in U for
α > 0,
3α 2 −
2α
√
Δ∗
√
3α 2 Δ∗
≤ nb ≤
,
2α
Δ∗ : 9α2 − 4α 4 .
4.20
5. Applications of Differential Inequalities
We apply the following result involving differential inequalities with a view to deriving
several further sufficient conditions for starlikeness of the n-fold symmetric function fb
defined by 1.4.
Lemma 5.1 Miller and Mocanu 13. Let Θu, v be a complex-valued function such that
Θ : D −→ C,
D ⊂ C × C,
5.1
C being (as usual) the complex plane, and let
u u1 iu2 ,
v v1 iv2 .
5.2
Suppose that the functions Θu, v satisfies each of the following conditions.
i Θu, v is continuous in D.
ii 1, 0 ∈ D and ReΘ1, 0 > 0.
iii ReΘiu2 , v1 ≤ 0 for all iu2 , v1 ∈ D such that
v1 ≤ −
1
1 u22 .
2
5.3
Let
pz 1 p1 z p2 z2 · · ·
5.4
pz, zp z ∈ D z ∈ U.
5.5
be analytic (regular) in U such that
International Journal of Mathematics and Mathematical Sciences
9
If
Re Θ pz, zp z ∈ D z ∈ U,
5.6
Re pz > 0 z ∈ U.
5.7
then
Let us now consider the following implication.
Theorem 5.2. Let n-fold symmetric function fb , defined by 1.4 and analytic in U with
f b z/z, / 0, z ∈ U, satisfy the following inequality:
zfb z
zfb z
zfb z
αnb
αλnb
nb
1−
−
,
> 1−
1 − α α1 − λ
αλ 1 Re
fb z
fb z
2
2
4
fb z
5.8
then
Re
zfb z
fb z
μ > 0,
nb
αnb
αλnb
z ∈ U; 1 −
1−
−
< 1; α ≥ 0, λ > 0; μ ≥ 1 .
2
2
4
5.9
Proof. If we put
pz zfb z
fb z
μ
,
5.10
then 5.8 is equivalent to
Re
αλ pz
μ
1−μ/μ
zp z α pz
2/μ
1 − αpz1/μ
nb
αnb
λαnb
− 1−
1−
>0
2
2
4
⇒ Re pz > 0
5.11
z ∈ U.
By setting pz u and zp z v and letting
αnb
λαnb
nb
αλ 1−μ/μ
2/μ
1/μ
u
1−
,
v αu 1 − αu − 1 −
Θz μ
2
2
4
5.12
10
International Journal of Mathematics and Mathematical Sciences
for α ≥ 0 and μ ≥ 1, we have the following.
i Θu, v is continuous in D C \ {0} × C.
ii 1, 0 ∈ D and
αλnb αnb nb αn2 b2
−
> 0,
4
2
2
4
ReΘ1, 0 5.13
since
nb
αnb
λαnb
1−
< 1.
1−
−
2
2
4
5.14
Thus, the conditions i and ii of Lemma 5.1 are satisfied. Moreover, for iu2 , v1 ∈ D and
v1 ≤ −1/21 u22 , we obtain
αλ
ReΘiu2 , v1 |u2 |1−μ/μ v1 cos
μ
1−μ π
π
α|u2 |2/μ cos
2μ
μ
1 − α|u2 |
1/μ
π
cos
2μ
nb
− 1−
2
αnb
1−
2
λαnb
4
π
π
αλ 1−μ/μ
2/μ
2
1 u2 |u2 |
α|u2 | cos
sin
≤−
2μ
2μ
μ
1 − α|u2 |
1/μ
π
cos
2μ
nb
− 1−
2
αnb
1−
2
5.15
λαnb
,
4
which, upon putting |u2 | ζ ζ > 0, yields
ReΘiu2 , v1 ≤ Φζ,
5.16
where
Φζ : −
π
π
π
αλ 1 ζ2 ζ1−μ/μ sin
αζ2/μ cos
1 − αζ1/μ cos
2μ
2μ
μ
2μ
nb
− 1−
2
αnb
1−
2
λαnb
.
4
5.17
Remark 5.3. If, for some choices of the parameters α, λ, μ, and nb, we find that
Φζ ≤ 0 ζ > 0,
5.18
International Journal of Mathematics and Mathematical Sciences
11
then we can conclude from 5.16 and Lemma 5.1 that the corresponding implication 5.8
holds true.
First of all, for the choice: μ 1 and nb 2, we have the following.
Theorem 5.4. If n-fold symmetric function fb , defined by 1.4 and analytic in U with
fb z
/ 0,
z
z ∈ U,
5.19
satisfies the following inequality:
zfb z
zfb z
zfb z
αλ
1 − α α1 − λ
αλ 1 >− ,
Re
fb z
fb z
2
fb z
5.20
then fb ∈ S∗ for any real α ≥ 0 and λ > 0.
Proof. For μ 1, nb 2, we find from 5.17 that
1
2
Φζ : −αλ
s − αs2 ≤ 0,
2
ζ ∈ R,
5.21
which implies Theorem 5.4 in view of the remark.
Remark 5.5. For λ 1, we will obtain the results by Kamali and Srivastava 12.
Acknowledgment
The work presented here was supported by UKM-ST-06-FRGS0244-2010.
References
1 P. T. Mocanu, “Une propriété de convexité généralisée dans la théorie de la représentation conforme,”
Mathematica, vol. 11, no. 34, pp. 127–133, 1969.
2 S. S. Miller, P. Mocanu, and M. O. Reade, “All α-convex functions are univalent and starlike,”
Proceedings of the American Mathematical Society, vol. 37, pp. 553–554, 1973.
3 K. Sakaguchi and S. Fukui, “On alpha-starlike functions and related functions,” Bulletin of Nara
University of Education, vol. 28, no. 2, pp. 5–12, 1979.
4 C. Ramesha, S. Kumar, and K. S. Padmanabhan, “A sufficient condition for starlikeness,” Chinese
Journal of Mathematics, vol. 23, no. 2, pp. 167–171, 1995.
5 S. Siregar, M. Darus, and B. A. Frasin, “Subordination and superordination for certain analytic
functions,” Bulletin of Mathematical Analysis and Applications, vol. 2, no. 2, pp. 42–49, 2010.
6 D. Bansal and R. K. Raina, “Some sufficient conditions for starlikeness using subordination criteria,”
Bulletin of Mathematical Analysis and Applications, vol. 2, no. 4, pp. 1–6, 2010.
7 S. Siregar, M. Darus, and T. Bulboacă, “A class of superordination-preserving convex integral
operator,” Mathematical Communications, vol. 14, no. 2, pp. 379–390, 2009.
8 B. A. Frasin and M. Darus, “Subordination results on subclasses concerning Sakaguchi functions,”
Journal of Inequalities and Applications, vol. 2009, Article ID 574014, 7 pages, 2009.
9 S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, vol. 225 of Monographs
and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.
12
International Journal of Mathematics and Mathematical Sciences
10 I. S. Jack, “Functions starlike and convex of order α,” Journal of the London Mathematical Society, vol. 3,
pp. 469–474, 1971.
11 S. Fukui, S. Owa, and K. Sakaguchi, “Some properties of analytic functions of Koebe type,” in Current
topics in Analytic Function Theory, pp. 106–117, World Scientific Publishing Company, New Jersey, NJ,
USA, 1992.
12 M. Kamali and H. M. Srivastava, “A sufficient condition for starlikeness of analytic functions of Koebe
type,” Journal of Inequalities in Pure and Applied Mathematics, vol. 5, no. 3, pp. 1–8, 2004.
13 S. S. Miller and P. T. Mocanu, “Second-order differential inequalities in the complex plane,” Journal of
Mathematical Analysis and Applications, vol. 65, no. 2, pp. 289–305, 1978.
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