volume 4, issue 2, article 36, 2003.
Received 16 December, 2002;
accepted 8 May, 2003.
Communicated by:A. Sofo
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Journal of Inequalities in Pure and Applied Mathematics
A CRITERION FOR p-VALENTLY STARLIKENESS
MUHAMMET KAMALI
Ataturk University,
Faculty of Science and Arts, Department of Mathematics, 25240, Erzurum-TURKEY.
E-Mail:mkamali@atauni.edu.tr
c
2000Victoria University ISSN (electronic): 1443-5756 148-02
A Criterion forp-Valently Starlikeness Muhammet Kamali
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Abstract
It is the purpose of the present paper to obtain some sufficient conditions for p- valently starlikeness for a certain class of functions which are analytic in the open unit diskE.
2000 Mathematics Subject Classification:30C45, 31A05.
Key words:p−valently starlikeness, Jack Lemma.
Contents
1 Introduction. . . 3 2 Preliminaries . . . 6 3 A Criterion forp-Valently Starlikeness. . . 9
A Criterion forp-Valently Starlikeness Muhammet Kamali
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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 by S(p)the subclass of A(p) consisting of functions which are p- 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 inE if it is regular, f(z)
z f0(z)6= 0, and
(1.2) Re
α
1 +zf00(z) f0(z)
+(1−α)zf0(z) f(z)
>0
A Criterion forp-Valently Starlikeness Muhammet Kamali
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for all z inE. The set of all such functions is denoted by α−CV, where α is a real number. Of course, if α = 1, then anα−convex function is convex;
and if α = 0, anα−convex function is starlike. Thus the sets α−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), namelyf(z) ≡ zp.If−1 < k 6 0,thenf(z)isp-valent convex; and if 0< k, thenf(z)isp-valent starlike. We can pass from (1.3) back to (1.2) if we divide by 1 +k > 0and set α = 1+k1 [6]. We denote byS(p, k)the subclass A(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:
A Criterion forp-Valently Starlikeness Muhammet Kamali
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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 Theorem1.1by 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).
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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. Let w(z) be analytic in E with w(0) = 0. If |w(z)| attains its maximum value in the circle |z| = r < 1 at a point z0, then we can write z0w0(z0) =kw(z0),wherekis a real number andk≥1.
Making use of Lemma2.1,we first prove
Lemma 2.2. Letq(z)be analytic inE withq(0) =pand suppose that
(2.1) Re
zq0(z) [q(z)]2
< 1
p(λ+ 1) (z ∈E,06λ 61), 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)
, where06λ 61.
Thenw(z) is analytic inE with w(0) = 0 and 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 .
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If we suppose that there exists a point z0 ∈ E such thatmax|z|6|z0||w(z)| =
|w(z0)|= 1,then, from Lemma2.1, 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
.
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This contradicts (2.1). Therefore, we have|w(z)|< 1inE, and it follows that Re{q(z)}>0inE. This completes our proof of Lemma2.2.
If we takeλ = 1in Lemma2.2, then we have the following Lemma2.3by Nunokawa [5].
Lemma 2.3. Letq(z)be analytic inE withq(0) =pand suppose that Re
zq0(z) [q(z)]2
< 1
2p (z ∈E). ThenRe{q(z)}>0inE.
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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 +z
f00(z)
f0(z) +kff0(z)(z) i0
h
1 +zf00(z)
f0(z) +kff(z)0(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 inEwithq(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
+kff0(z)(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) +kzff0(z)(z) −1 1 +zff000(z)(z)+kzff(z)0(z) +z
kz
f0(z) f(z)
0
+z
f00(z) f0(z)
0
1 +zff000(z)(z) +kzff0(z)(z)
= 1 + z2
f00(z) f0(z)
0
+kf0(z) f(z)
0
−1 1 +zff000(z)(z) +kzff(z)0(z) ,
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or
(k+ 1)q(z) +zq0(z) q(z)
= 1 + z2
f00(z) f0(z)
0
+kf0(z) f(z)
0
−1
1 +zff000(z)(z)+kzff0(z)(z) + (k+ 1)q(z)
= 1 + z2
f00(z)
f0(z) +kff(z)0(z)0
+ 2z
f00(z)
f0(z) +kff(z)0(z)
+z2
f00(z)
f0(z) +kff0(z)(z)2
1 +zff000(z)(z) +kzff0(z)(z)
= 1 +z
f00(z)
f0(z) +kf0(z) f(z)
+z
z
f00(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) +kff(z)0(z)
1 +zff000(z)(z)+kzff0(z)(z) 2
= 1 +z h
1 +zf00(z)
f0(z) +kff0(z)(z)i0
1 +zff000(z)(z) +kzff(z)0(z)2 . From Lemma2.3and (3.1), we thus find that
Re
1 +zf00(z)
f0(z) +kzf0(z) f(z)
>0 (z ∈E, k >0).
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This completes our proof of Theorem3.1.
If we take α = 0, after writing k+11 = α in (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).
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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. MILLER AND 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 Theory, 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.