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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:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 148-02

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A Criterion forp-Valently Starlikeness Muhammet Kamali

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J. Ineq. Pure and Appl. Math. 4(2) Art. 36, 2003

http://jipam.vu.edu.au

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.

1. Introduction

LetA(p)be the class of functions of the form:

f(z) =z^p+

∞

X

n=p+1

a_nzⁿ (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

a_nzⁿ.

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

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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) = z^p+

∞

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) ≡ z^p.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 + ^zf_f0⁰⁰(z)^(z) zf⁰(z)

f(z)

,

we recall their result as:

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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(4e^p−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) = e^iθ,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 · λe^3iθ+ 2e^2iθ+λe^iθ (e^2iθ + 2λe^iθ+ 1)²

= 2k

p · λe^3iθ+ 2e^2iθ+λe^iθ

(e^2iθ+ 2λe^iθ+ 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) +k^f_f⁰_(z)^(z) i0

h

1 +z_f₀₀_(z)

f⁰(z) +k^f_f(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)

(k >0). Then,q(z)is analytic inEwithq(0) =p, q(z)6= 0inE.We have

q⁰(z) q(z) =

z^f_f⁰⁰0(z)^(z)

0

+

kz^f_f⁰_(z)^(z)0

1 +z^f_f⁰⁰0(z)^(z) +kz^f_f(z)⁰^(z) =

f⁰⁰(z) f⁰(z) +z

f⁰⁰(z) f⁰(z)

0

+k^f_f⁰_(z)^(z) +kz

f⁰(z) f(z)

0

1 +z^f_f⁰⁰0(z)^(z) +kz^f_f(z)⁰^(z) . Then, we obtain

zq⁰(z)

q(z) = 1 +z^f_f⁰⁰0(z)^(z) +kz^f_f⁰_(z)^(z) −1 1 +z^f_f⁰⁰0(z)^(z)+kz^f_f(z)⁰^(z) +z

kz

f⁰(z) f(z)

0

+z

f⁰⁰(z) f⁰(z)

0

1 +z^f_f⁰⁰0(z)^(z) +kz^f_f⁰_(z)^(z)

= 1 + z²

_f00(z) f⁰(z)

⁰

+k_f0(z) f(z)

⁰

−1 1 +z^f_f⁰⁰0(z)^(z) +kz^f_f(z)⁰^(z) ,

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or

(k+ 1)q(z) +zq⁰(z) q(z)

= 1 + z²

_f00(z) f⁰(z)

⁰

+k_f0(z) f(z)

⁰

−1

1 +z^f_f⁰⁰0(z)^(z)+kz^f_f⁰_(z)^(z) + (k+ 1)q(z)

= 1 + z²

f⁰⁰(z)

f⁰(z) +k^f_f(z)⁰^(z)0

+ 2z

f⁰⁰(z)

f⁰(z) +k^f_f(z)⁰^(z)

+z²

f⁰⁰(z)

f⁰(z) +k^f_f⁰_(z)^(z)2

1 +z^f_f⁰⁰0(z)^(z) +kz^f_f⁰_(z)^(z)

= 1 +z

f⁰⁰(z)

f⁰(z) +kf⁰(z) f(z)

+z

z

f⁰⁰(z)

f⁰(z) +k^f_f⁰_(z)^(z)0

+

f⁰⁰(z)

f⁰(z) +k^f_f(z)⁰^(z)

1 +z^f_f⁰⁰0(z)^(z)+kz^f_f(z)⁰^(z) . Thus,

1 + 1

k+ 1z q⁰(z)

[q(z)]² = 1 +z z

f⁰⁰(z)

f⁰(z) +k^f_f(z)⁰^(z)0

+

f⁰⁰(z)

f⁰(z) +k^f_f(z)⁰^(z)

1 +z^f_f⁰⁰0(z)^(z)+kz^f_f⁰_(z)^(z) 2

= 1 +z h

1 +z_f00(z)

f⁰(z) +k^f_f⁰_(z)^(z)i⁰

1 +z^f_f⁰⁰0(z)^(z) +kz^f_f(z)⁰^(z)2 . From Lemma2.3and (3.1), we thus find that

Re

1 +zf⁰⁰(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 + ^zf_f0⁰⁰(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.