On sufficient condition for starlikeness

On sufficient condition for starlikeness

D. O. Makinde, T. O. Opoola

Abstract

In this paper,we give a condition for starlikeness of the integral oper- ator of the formF(z) =

Z ^z

0 k

Y

ı=1

fı(s) s

α¹

ds.

2000 Mathematics Subject Classification:30C45

Key words and phrases: Starlike, Convex, Briot-Bouquet subordination.

1 Introduction

LetA be the class of all analytic functions f(z) defined in the open unit disk U ={z∈C :|z|<1}andSthe subclass ofAconsisting of univalent functions

(1) f(z) =z+

X∞ k=2

a_kz^k

S^∗={f ∈S:Re(zf⁰(z)

f(z) )>0, z∈U}, Mα={f ∈S : f(z)f⁰(z)

z 6= 0, ReJ(α, f;z)>0, z∈U}

1Received 14 February, 2009

Accepted for publication (in revised form) 17 March, 2009

35

where J(α, f;z) = (1−α)^zf_f(z)^0(z) +α(1 + ^zf_f0⁰⁰(z)^(z)) be the class of starlike and α−convexfunctions respectively.

Letp(z) be the class of functions that are regular in U and of the form :

(2) p(z) = 1 +

X∞ k=1

bkz^k Furthermore, let h(z) = ^1+z_1−z.

Let T be the univalent [5] subclass of Aconsisting of functionsf(z) satisfying

|^z_f(z)^2f0(z)₂ −1|<1,(z∈U)

LetT_n be the subclass ofT for which f^k(0) = 0 (k= 2,3, ..., n).

Let T_n,µ be the subclass of T_n consisting of functions of the form Rz

0

Qk

ı=1(^fı_s^(s))^α1dssatisfying: |^z_f(z)^2f0(z)₂ −1|< µ,(z∈U) for someµ(0< µ≤1).

2 Preliminaries

Theorem 1 [1] LetM andN be analytic inU withM(0) =N(0) = 0.IfN(z) maps onto a many sheeted region which is starlike with respect to the origin and Re{^M_N0⁰_(z)^(z)}>0 in U, then Re{^M_N(z)^(z)}>0 in U.

Theorem 2 [6] Letf_ı ∈T_n,µı(ı= 1,2, ..., k;k∈ N^∗) be defined by

(3) f_i(z) =z+

X∞ n=2

aⁱ_nzⁿ

for all i = 1,2, ..., k;α, β ∈ C;R{β} ≥ γ and γ = Xk i=1

1 + (1 +µ_i)M

|α| (M ≥ 1,0< µ_i <1, k∈N^∗). If |fi(z)| ≤ M(z∈U), i= 1,2, ..., k then, the integral operator

(4) F_α,β(z) ={β

Z z 0

t^β−1 Yk i=1

(f_i(t) t )^α1dt}^β1 is univalent.

Theorem 3 [2] Let h be convex in U and Re{βh(z) +γ} >0, z ∈U.If p ∈ H(U) whereH(U) is the class of functions which are analytic in the unit disk, with p(0) =h(0)andpsatisfies the Briot-Bouquet differential subordinations:

p(z) +_βp(z)+γ^zp0(z) ≺h(z), z∈U. Then, p(z)≺h(z), z∈U.

3 Main Results

We now give the proof of the following results:

Theorem 4 Let Fα(z) be the function in U defined by

(5) F_α(z) =

Z z 0

Yk i=1

(f_i(s)

s )^α1ds, α∈C.

If f_i ∈S^∗ then, F(z)∈S^∗ where f_i is as in equation (3) above.

Proof. By differentiating (5), we obtain: F⁰(z) = Qk

i=1(^fi(z)_z )^1α. Thus, zF⁰(z)

F(z) = Qk

i=1(^fi_z^(z))^α1 Rz

0

Qk

i=1(^fi(s)_s )^α1ds. Let

(6) M =zF⁰(z), N(z) =F(z)

From (5) and (6) we have:

M⁰(z)

N⁰(z) = 1 + zF⁰⁰(z)

F⁰(z) , M⁰(z) N⁰(z) = 1 +

Pk i=1 1

α(^zf_f(z)^i0(z)−1) Qk

i=1(^fi_z^(z))^α1

|M⁰(z)

N⁰(z) −1|= |Pk i=1 1

α(^zf_f(z)^i0(z)−1)|

|Qk

i=1(^fi_z^(z))^α1| ≤ Pk

i=1|_α¹||^zf_f(z)^i0(z)−1|

|Qk

i=1(^fi_z^(z))^α1| .

By hypothesis f_i ∈ S^∗. This means that |^zf_f(z)^i0(z)−1| <1, which implies that

|^M_N0⁰(z)^(z) −1|<1. Thus Re{^M_N0⁰(z)^(z)}>0 and by Theorem 1,Re{^M_N(z)^(z)}>0. This implies thatRe{^zF_F(z)^0(z)}>0. Hence F ∈S^∗.

Remark 1 The integral in (5) is equivalent to that in (4) of section 2 with β = 1.

Let S={f :U →C} ∩S. LetF(z)∈U be defined by

(7) F(z) =

Z z 0

Yk i=1

(f_i(s) s )^α1ds.

Theorem 5 Letz∈U, α∈C, Reα >0andm_α=M_α∩s.IfF ∈m_α, then F ∈ S^∗that is m_α⊂S^∗.

Proof. From (6) above, we have ^F(z)F_z^0(z) 6= 0 and forF ∈m_α,we have (8) ReJ(α, f;z) =Re{(1−α)zF⁰(z)

F(z) +α(1 + zF⁰(z) F(z) )}

forp(z) = ^zF_F(z)^0(z), ^zp_p(z)^0(z) = 1 +^zF_F0^”(z)_(z) −p(z).This implies that

(9) 1 +zF⁰⁰(z)

F⁰(z) = zp⁰(z)

p(z) +p(z) using (7) and (9) in (8), we obtain

(10) ReJ(α, f;z) =Re{(1−α)p(z) +α(zp⁰(z)

p(z) +p(z))}.

Simplifying (10), we obtainReJ(α, f;z) =Re{p(z) +α(^zp_p(z)^0(z))}

p(0) + ^αzp_p(0)⁰⁽⁰⁾ = 1 and p(0) = h(0) = 1. Thus, using Theorem 3 with β = 1 andγ = 0,we havep(z)+^αzp_p(z)^0(z) < h(z) = ^1+z_1−z. This implies thatp(z)≺h(z).

That isRe{p(z)}>0. Thus, Re{^zF_F(z)^0(z)}>0. Hence, F ∈S^∗.

References

[1] S.A. Halim,On a class of Analytic Functions involing, Tamkang, Journal of Mathematics vol.23 Number 1, Spring 1992.

[2] M. Acu and S. Owa, On some subclasses of univalent function, Journal of inequalities in pure and applied mathematics, vol. 6,(3) art 70, 2005.

[3] S. Kanas and F. Ronning, Uniformly Starlike and convex functions and other related classes of univalent functions, Ann University, Marie Curie- Sklodowska Section A, 53, 1999, 95-105.

[4] S.S. Miller and P.T. Mocanu, Univalent solutions of Briot-Bouquet dif- ferential equations, Journal of Differential equations, 56, 1985, 297-309.

[5] S. Ozaki and M.Nunokawa, The Schwarzian derivative and Univalent functions,, Pro. Ameri.Math. Soc., 33, 1972, 302-394.

[6] N. Seenivasagan, Sufficient Conditions for Univalence, Applied Mathe- matics E-Notes, 8, 2008, 30-35.

D. O. Makinde

Department of Mathematics Obafemi Awolowo University Ile Ife 220005, Nigeria

e-mail: [email protected], [email protected] Timothy O. Opoola

University of Ilorin Mathematics

Department of Mathematics, University of Ilorin, Ilorin,Nigeria e-mail: [email protected]