http://jipam.vu.edu.au/
Volume 4, Issue 2, Article 36, 2003
A CRITERION FOR p-VALENTLY STARLIKENESS
MUHAMMET KAMALI ATATURKUNIVERSITY, FACULTY OFSCIENCE ANDARTS, DEPARTMENT OFMATHEMATICS,
25240, ERZURUM-TURKEY.
mkamali@atauni.edu.tr
Received 16 December, 2002; accepted 8 May, 2003 Communicated by A. Sofo
ABSTRACT. It is the purpose of the present paper to obtain some sufficient conditions forp- valently starlikeness for a certain class of functions which are analytic in the open unit diskE.
Key words and phrases: p−valently starlikeness, Jack Lemma.
2000 Mathematics Subject Classification. 30C45, 31A05.
1. INTRODUCTION
LetA(p)be the class of functions of the form:
f(z) =zp +
∞
X
n=p+1
anzn (p∈N={1,2,3, . . .}), which are analytic inE ={z ∈C:|z|<1}.
A functionf(z)∈A(p)is said to be p-valently starlike if and only if Re
zf0(z)
f(z)
>0 (z ∈E).
We denote byS(p)the subclass ofA(p)consisting of functions which arep-valently inE (see, e.g., Goodman [1]).
Let
(1.1) f(z) = z+
∞
X
n=2
anzn.
A functionf(z)of the form (1.1) is said to beα−convex inEif it is regular, f(z)
z f0(z)6= 0,
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
148-02
and
(1.2) Re
α
1 +zf00(z) f0(z)
+(1−α)zf0(z) f(z)
>0
for allzinE. The set of all such functions is denoted byα−CV, whereαis a real number. Of course, ifα = 1, then an α−convex function is convex; and ifα= 0, anα−convex function is starlike. Thus the setsα−CV give a “continuous” passage from convex functions to starlike functions. Sakaguchi considers functions of the form
f(z) =zp+
∞
X
n=p+1
anzn,
wherepis a positive integer, and he imposes the condition
(1.3) Re
1 + zf00(z)
f0(z) +kzf0(z) f(z)
>0
for z inE. He proved that if k = −1, there is only one function that satisfies (1.3), namely f(z) ≡ zp. If−1 < k 6 0,thenf(z)is p-valent convex; and if 0 < k, then f(z) isp-valent starlike. We can pass from (1.3) back to (1.2) if we divide by1 +k >0and setα= 1+k1 [6]. We denote byS(p, k)the subclassA(p)consisting of functions which satisfy the condition (1.3).
Obradovic and Owa [7] have obtained a sufficient condition for starlikeness off(z)∈ A(1) which satisfies a certain condition for the modulus of
1 + zff000(z)(z) zf0(z)
f(z)
,
we recall their result as:
Theorem 1.1. Iff(z)∈A(1)satisfies
1 + zf00(z) f0(z)
< K
zf0(z) f(z)
(z ∈E), whereK = 1.2849...,thenf(z)∈S(1).
Nunokawa [4] improved Theorem 1.1 by proving Theorem 1.2. Iff(z)∈A(p), and if
1 + zf00(z) f0(z)
<
zf0(z) f(z)
1
plog(4ep−1) (z ∈E), thenf(z)∈S(p).
2. PRELIMINARIES
In order to obtain our main result, we need the following lemma attributed to Jack [2] (given also by Miller and Mocanu [3]).
Lemma 2.1. Letw(z)be analytic inE withw(0) = 0.If|w(z)|attains its maximum value in the circle|z| = r < 1at a pointz0, then we can write z0w0(z0) = kw(z0),where k is a real number andk≥1.
Making use of Lemma 2.1,we first prove
Lemma 2.2. Letq(z)be analytic inEwithq(0) =pand suppose that
(2.1) Re
zq0(z) [q(z)]2
< 1
p(λ+ 1) (z ∈E,06λ61), thenRe{q(z)}>0inE.
Proof. Let us put
q(z) = p 1
2 +1 2λ
1 +w(z) 1−w(z)+
1 2− 1
2λ
1−w(z) 1 +w(z)
, where06λ61.
Thenw(z)is analytic inEwithw(0) = 0and by an easy calculation, we have 1 +z q0(z)
[q(z)]2 = 1 + 2
p· (λw2(z) + 2w(z) +λ)zw0(z) (w2(z) + 2λw(z) + 1)2 .
If we suppose that there exists a pointz0 ∈E such thatmax|z|6|z0||w(z)| =|w(z0)| = 1,then, from Lemma 2.1, we havez0w0(z0) =kw(z0), (k >1).
Puttingw(z0) = eiθ,we find that z0 q0(z0)
[q(z0)]2 = 2
p ·λw2(z0)w0(z0)z0+ 2w(z0)w0(z0)z0+λw0(z0)z0 [w2(z0) + 2λw(z0) + 1]2
= 2k
p · λe3iθ+ 2e2iθ+λeiθ (e2iθ+ 2λeiθ+ 1)2
= 2k
p · λe3iθ+ 2e2iθ +λeiθ
(e2iθ+ 2λeiθ+ 1)2 · e−2iθ + 2λe−iθ + 12
(e−2iθ + 2λe−iθ + 1)2
= k
p · λcos 3θ+ (4λ2+ 2) cos 2θ+ (11λ+ 4λ3) cosθ+ (8λ2+ 2) 4 (λ+ cosθ)4
= k
p · (1 +λcosθ) (λ+ cosθ)2 (λ+ cosθ)4
= k
p · 1 +λcosθ (λ+ cosθ)2, so that
Re
z0 q0(z0) [q(z0)]2
= k
p · 1 +λcosθ (λ+ cosθ)2 = k
p · λ2+λcosθ+ 1−λ2 (λ+ cosθ)2
= k p
λ
(λ+ cosθ) + 1−λ2 (λ+ cosθ)2
> 1 p
1 λ+ 1
.
This contradicts (2.1). Therefore, we have|w(z)| <1inE, and it follows that Re{q(z)} > 0
inE. This completes our proof of Lemma 2.2.
If we takeλ= 1in Lemma 2.2, then we have the following Lemma 2.3 by Nunokawa [5].
Lemma 2.3. Letq(z)be analytic inEwithq(0) =pand suppose that Re
zq0(z) [q(z)]2
< 1
2p (z ∈E).
ThenRe{q(z)}>0inE.
3. A CRITERION FOR p-VALENTLY STARLIKENESS
Theorem 3.1. Letf(z)∈A(p), f(z)6= 0,in0<|z|<1and suppose that
(3.1) Re
1 +z
h
1 +zf00(z)
f0(z) +kff(z)0(z)i0 h
1 +z
f00(z)
f0(z) +kff0(z)(z)i2
<1 + 1 k+ 1
1 2p
(z ∈E).
Thenf(z)∈S(p, k).
Proof. Let us put
q(z) = 1 k+ 1
1 +zf00(z)
f0(z) +kzf0(z) f(z)
(k > 0).
Then,q(z)is analytic inE withq(0) =p, q(z)6= 0inE.We have
q0(z) q(z) =
zff000(z)(z)
0
+
kzff0(z)(z)0
1 +zff000(z)(z) +kzff(z)0(z) =
f00(z) f0(z) +z
f00(z) f0(z)
0
+kff(z)0(z) +kz
f0(z) f(z)
0
1 +zff000(z)(z) +kzff(z)0(z) . Then, we obtain
zq0(z)
q(z) = 1 +zff000(z)(z)+kzff(z)0(z) −1 1 +zff000(z)(z) +kzff(z)0(z) +z
kzf0(z) f(z)
0
+zf00(z) f0(z)
0 1 +zff000(z)(z)+kzff0(z)(z)
= 1 + z2
f00(z) f0(z)
0
+k
f0(z) f(z)
0
−1 1 +zff000(z)(z) +kzff0(z)(z) , or
(k+ 1)q(z) +zq0(z) q(z)
= 1 + z2
f00(z) f0(z)
0
+k
f0(z) f(z)
0
−1
1 +zff000(z)(z) +kzff(z)0(z) + (k+ 1)q(z)
= 1 + z2
f00(z)
f0(z) +kff(z)0(z)0
+ 2z
f00(z)
f0(z) +kff0(z)(z)
+z2
f00(z)
f0(z) +kff(z)0(z)2
1 +zff000(z)(z)+kzff0(z)(z)
= 1 +z
f00(z)
f0(z) +kf0(z) f(z)
+z
zf00(z)
f0(z) +kff0(z)(z)0
+f00(z)
f0(z) +kff(z)0(z)
1 +zff000(z)(z) +kzff(z)0(z) .
Thus,
1 + 1
k+ 1z q0(z)
[q(z)]2 = 1 +z z
f00(z)
f0(z) +kff(z)0(z)0
+
f00(z)
f0(z) +kff0(z)(z)
1 +zff000(z)(z) +kzff(z)0(z)2
= 1 +z h
1 +z
f00(z)
f0(z) +kff(z)0(z)i0
1 +zff000(z)(z) +kzff0(z)(z) 2 . From Lemma 2.3 and (3.1), we thus find that
Re
1 +zf00(z)
f0(z) +kzf0(z) f(z)
>0 (z ∈E, k > 0).
This completes our proof of Theorem 3.1.
If we takeα= 0, after writing k+11 =αin (3.1), then we obtain M. Nunokawa’s theorem as follows.
Theorem 3.2. Letf(z)∈A(p), f(z)6= 0,in0<|z|<1and suppose that Re
(1 + zff000(z)(z) zf0(z)
f(z)
)
<1 + 1
2p, z ∈E.
Thenf(z)∈S(p).
REFERENCES
[1] A.W. GOODMAN, On the Schwarz-Christoffel transformation andp-valent functions, Trans. Amer.
Math. Soc., 68 (1950), 204–223.
[2] I.S. JACK, Functions starlike and convex of orderα, J. London Math. Soc., 2(3) (1971), 469–474.
[3] S.S. MILLERAND P.T. MOCANU, Second order differential inequalities in the complex plane, J.
Math. Anal. Appl., 65 (1978), 289–305.
[4] M. NUNOKAWA, On certain multivalent functions, Math. Japon., 36 (1991), 67–70.
[5] M. NUNOKAWA, A certain class of starlike functions, in Current Topics in Analytic Function The- ory, H.M. Srivastava and S. Owa (Eds.), Singapore, New Jersey, London, Hong Kong, 1992, p.
206–211.
[6] A.W. GOODMAN, Univalent Functions, Volume I, Florida, 1983, p.142–143.
[7] M. OBRADOVICANDS. OWA, A criterion for starlikeness, Math. Nachr., 140 (1989), 97–102.