Z z 0 f(u) u α1 du (1.1) for some complex numbers α(α6= 0)

UNIVALENCE CRITERION FOR AN INTEGRAL OPERATOR

Virgil Pescar, Daniel Breaz

Abstract.We consider the integral operator denoted by T_α,β and for the func- tionf ∈ Awe proved a sufficient condition for univalence from this integral operator.

2000 Mathematics Subject Classification: 30C45.

Key Words and Phrases: Integral operator, univalence, starlike.

1.Introduction LetA be the class of functionsf of the form

f(z) =z+

∞

X

n=2

anzⁿ

which are analytic in the open unit disk U = {z∈C:|z|<1}. Let S denote the subclass of Aconsisting of all univalent functions f inU.

Forf ∈ A, the integral operator Gα is defined by G_α(z) =

Z z

0

f(u) u

_α¹

du (1.1)

for some complex numbers α(α6= 0).

In [1] Kim-Merkes prove that the integral operatorGαis in the classSfor _|α|¹ ≤ ¹₄ and f ∈ S.

Also, the integral operatorJ_γ forf ∈ Ais given by M_γ(z) =

1 γ

Z _z

0

u⁻¹(f(u))^γ1 du γ

(1.2) γ be a complex number,γ 6= 0.

Miller and Mocanu [3] have studied that the integral operatorMγ is in the class S for f ∈ S^∗, γ >0,S^∗ is the subclass of S consisting of all starlike functionsf in U.

We consider the integral operator T_α,β defined by

T_α,β =

"

β Z z

0

u^β−1 f(u)

u _α¹

du

#_β¹

(1.3) forf ∈ Aand α, β be complex numbers,α6= 0,β 6= 0.

We need the following lemmas.

Lemma 1.1.[6].Let α be a complex number, Re α >0 and f ∈ A. If 1− |z|^{2Re α}

Re α

zf⁰⁰(z) f⁰(z)

≤1 (1.4)

for all z∈ U, then for any complex number β, Re β ≥Re α the function

Fβ(z) =

β Z z

0

u^β−1f⁰(u)du _β¹

(1.5) is in the class S.

Lemma 1.2.(Schwarz [2]). Let f the function regular in the disk

U_R = {z∈C:|z|< R} with |f(z)| < M, M fixed. If f(z) has in z = 0 one zero with multiply ≥m, then

|f(z)| ≤ M

R^m|z|^m, z ∈ U_R (1.6)

the equality (in the inequality (1.6) for z6= 0) can hold only if f(z) =e^iθ M

R^mz^m, where θ is constant.

2.Main results

Theorem 2.1.Let α be a complex number,a=Re _α¹ >0 and f ∈ A, f(z) =z+a₂z²+. . ..

If

zf⁰(z) f(z) −1

≤ (2a+ 1)^2a+12a

2 |α| (2.1)

for all z∈ U, then for any complex number β, Re β ≥Re _α¹, the function

T_α,β(z) =

"

β Z z

0

u^β−1 f(u)

u _α¹

du

#_β¹

(2.2) is in the class S.

Proof. Let us consider the function g(z) =

Z z

0

f(u) u

¹

α

du (2.3)

The function f is regular inU. From (2.3) we have g⁰(z) =

f(z) z

_α¹ ,

g⁰⁰(z) = 1 α

f(z) z

_α¹−1

zf⁰(z)−f(z) z²

We define the function h(z) = ^zg_g0⁰⁰(z)^(z), z∈ U and we obtain h(z) = zg⁰⁰(z)

g⁰(z) = 1 α

zf⁰(z) f(z) −1

, z∈ U (2.4)

The function hsatisfies the condition h(0) = 0. From (2.1) and (2.4) we have

|h(z)| ≤ (2a+ 1)^2a+12a

2 (2.5)

for all z∈ U. Applying Lemma 1.2 we get

|h(z)| ≤ (2a+ 1)^2a+12a

2 |z| (2.6)

for all z∈ U.

From (2.4) and (2.6) we obtain 1− |z|^2a

a

zg⁰⁰(z) g⁰(z)

≤ (2a+ 1)^2a+12a 2

1− |z|^2a

a |z| (2.7)

for all z∈ U.

Because

max|z|≤1

1− |z|^2a a |z|

= 2

(2a+ 1)^2a+12a from (2.7) we have

1− |z|^2a a

zg⁰⁰(z) g⁰(z)

≤1 (2.8)

for all z∈ U.

From (2.3) we have g⁰(z) = _f(z)

z

_α¹

, and by Lemma 1.1 we obtain that the integral operator T_α,β define by (2.2) is in the classS.

Corollary. 2.2.Let α be a complex number, a=Re _α¹ >0 andf ∈ A, f(z) =z+a₂z²+a₃z³+. . .

If

zf⁰(z) f(z) −1

≤ (2a+ 1)^2a+12a

2 |α| (2.9)

for all z∈ U, then the integral operator M_α given by (1.2) belongs to class S.

Proof. For β= _α¹, from Theorem 2.1 we obtain Corollary 2.2.

Corollary. 2.3.Let α be a complex number, a = Re _α¹ ∈ (0,1] and f ∈ A, f(z) =z+a₂z²+. . ..

If

zf⁰(z) f(z) −1

≤ (2a+ 1)^2a+12a

2 |α| (2.10)

for all z∈ U, then the integral operator Gα, is in the class S.

Proof. We take β = 1 in Theorem 2.1.

References

[1] Y. J. Kim, E. P. Merkes, On an Integral of Powers of a Spirallike Function, Kyungpook Math. J., 12 (1972), 249-253.

[2] O. Mayer, The Functions Theory of One Variable Complex, Bucure¸sti, 1981.

[3] S. S. Miller, P. T. Mocanu,Differential Subordinations, Theory and Applica- tions, Monographs and Text Books in Pure and Applied Mathematics, 225, Marcel Dekker, New York, 2000.

[4] P. T. Mocanu, T. Bulboac˘a, G. St. S˘al˘agean, The Geometric Theory of Univalent Functions, Cluj, 1999.

[5] Z. Nehari,Conformal Mapping, Mc Graw-Hill Book Comp., New York, 1952 (Dover. Publ. Inc., 1975).

[6] N. N. Pascu,An Improvement of Becker’s Univalence Criterion, Proceedings of the Commemorative Session Simion Stoilow, University of Bra¸sov, 1987, 43-48.

[7] N.N. Pascu and V. Pescar, On the Integral Operators of Kim-Merkes and Pfaltzgraff, Mathematica, Univ. Babe¸s-Bolyai, Cluj-Napoca, 32 (55), 2 (1990), 185- 192.

Virgil Pescar

Department of Mathematics

”Transilvania” University of Bra¸sov 500091 Bra¸sov, Romania

e-mail: [email protected] Daniel Breaz

Department of Mathematics

”1 Decembrie 1918” University of Alba Iulia 510009 Alba Iulia, Romania

e-mail: [email protected]