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

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Volume 4, Issue 2, Article 36, 2003

A CRITERION FOR p-VALENTLY STARLIKENESS

MUHAMMET KAMALI ATATURKUNIVERSITY, FACULTY OFSCIENCE ANDARTS, DEPARTMENT OFMATHEMATICS,

25240, ERZURUM-TURKEY.

[email protected]

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) =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 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

a_nzⁿ.

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

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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) =z^p+

∞

X

n=p+1

a_nzⁿ,

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) ≡ z^p. 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 + ^zf_f0⁰⁰(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(4e^p−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) = 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

.

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 +z_f00(z)

f⁰(z) +k^f_f(z)^0(z)i⁰ h

1 +z

f⁰⁰(z)

f⁰(z) +k^f_f⁰_(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) =

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_f(z)^0(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_f(z)^0(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_f⁰_(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 +z^f_f⁰⁰0(z)^(z) +kz^f_f(z)^0(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_f⁰_(z)^(z)

+z²

f⁰⁰(z)

f⁰(z) +k^f_f(z)^0(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_f⁰_(z)^(z)

1 +z^f_f⁰⁰0(z)^(z) +kz^f_f(z)^0(z)2

= 1 +z h

1 +z

f⁰⁰(z)

f⁰(z) +k^f_f(z)^0(z)i0

1 +z^f_f⁰⁰0(z)^(z) +kz^f_f⁰_(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 + ^zf_f0⁰⁰(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.