Holhos Viorel. 85KB Jun 04 2011 12:08:46 AM

ACTA UNIVERSITATIS APULENSIS No 9/2005

ON SOME CONDITIONS FOR n-STARLIKENESS

Viorel Cosmin Holhos

Abstract. _{In this paper we obtain a sufficient condition for n-starlikeness} of the form: (2α−1)D_Dn+1n_ff(z)

(z) −1

+ (1−α)D_Dn+2n f(z)

f(z) ≺ h(z)where h(z) is

an univalent function in the unit disc U and Dn is the S˘al˘agean differential operator.

1. Introduction

Let An, n ∈ N∗ denote the class of functions of the form: f(z) = z +

∞

P

k=n+1

akzkwich are analytic in the unit discU ={z;z ∈C, |z|<1}andA1 =

A.

We note S∗

= nf ∈ A: Rezf_f(z)′(z) >0, z∈Uothe class of functions f ∈ A which are starlike in the unit disc.

We denote by K the class of functions f ∈ Awhich are convex in the unit disc U, that is K=nf ∈ A : Rezf_f′′′_(z)(z) + 1 >0, z ∈U

o

.

For f ∈ An we define the S˘al˘agean differential operatorDn ([2]) by

D0f(z) = f(z)

D1f(z) = Df(z) = zf′

(z) and Dn+1_{f(z) =D(D}n_{f(z)); n∈N}∗

∪ {0}.

Letα ∈[0,1) and letn ∈N. The classSn(α) named the class ofn−starlike

function of order α is defined by Sn(α) =

n

f ∈ A: ReD_Dn+1n f(z)

f(z) > α, z ∈U o

. Theorem 1._{[1]Let qbe a univalent function in Uand let the functions θ, φ}

be analytic in a domain D containing q(U), with φ(w) 6= 0, when w∈ q(U).

Set Q(z) = zq′

(z)φ(q(z)), h(z) =θ(q(z)) +Q(z) and suppose that

Viorel Cosmin Holhos - On some conditions for n-starlikeness

2. Main results

Theorem 2._{Let α ∈ [0,1], n ∈ N , f(z) ∈ A and let q be a convex}

and q is the best dominant of (2).

Proof. For α= 1 it is evident. Suppose that 0≤ α <1. In Theorem 1 we

choose

θ(w) = (1−α)w2+ (2α−1)w−α φ(w) = 1−α

and we have in (i)Q(z) = (1−α)zq′

(z) is starlike in U, becauseq is convex.

Viorel Cosmin Holhos - On some conditions for n-starlikeness which implies that

Dn+1_f(z)

Dn_f(z) ≺q(z)

Remark._{Forn = 0 we obtain the result given in [3].}

Corollary 1. _{Letα ∈[0,1] and let f(z)∈ A, that satisfy}

Viorel Cosmin Holhos - On some conditions for n-starlikeness

Corollary 2. _{Iff(z)∈ A satisfies} Re

"

Dn+2f(z) Dn_f(z)

#

>−1

2, z ∈U, (6)

then

f ∈S∗

n 1

2

. (7)

References

[1]. S. S. Miller, P. T. Mocanu, On some classes of first-order differential

subordinations, Michigan Math. J., 32(1985), 185-195.

[2]. G. S. S˘al˘agean, Subclasses of univalent functions, Lecture Notes in Math, Springer-Verlag, 1013(1983), 362-372.

[3]. M. Obradovi´c, T. Yaguchi, H. Saitoh, On some conditions for

univa-lence and starlikeness in the unit disc, Rendiconti di Matematica, Serie VII,

Volume 12, Roma (1992), 869-877. C. Holhos:

Department of Mathematics Hunedoara

Romania

email:ancaholhos@yahoo.com