2 Sufficient Conditions For Univalence

Certain sufficient conditions for univalence

Narayanasamy Seenivasagan and Daniel Breaz

Abstract

In this paper, we determined conditions on β, α_i and f_i(z) so that the integral operator

( β

Z _z

0

t^β−1 Yn i=1

(f_i(t) t )^αi1 dt

)¹

β

is univalent in the open unit disk for the two subclasses analytic functions.

2000 Mathematical Subject Classification: Primary 30C80, Secondary 30C45.

Key words: Integral operator, univalent functions, Schwartz’s Lemma.

1 Introduction

Let A be the class of all analytic functions f(z) defined in the open unit disk U := {z ∈ C: |z| < 1} and normalized by the conditions f(0) = 0 = f⁰(0)−1. Let Sbe the subclass ofAconsisting of univalent functions in U.

Let A2 be the subclass of Aconsisting of functions is of the form

(1.1) f(z) = z+

X∞

k=3

a_kz^k.

1Received 12 September, 2007

Accepted for publication (in revised form) 15 October, 2007

7

Let T be the univalent [6] subclass of A which satisfies (1.2)

¯¯

¯¯z²f⁰(z) (f(z))² −1

¯¯

¯¯<1 (z ∈U).

LetT₂ be the subclass of T for whichf⁰⁰(0) = 0. Let T_2,µ be the subclass of T₂ consisting of functions is of the form (1.1) which satisfy

(1.3)

¯¯

¯¯z²f⁰(z) (f(z))² −1

¯¯

¯¯≤µ (z ∈U)

for some µ (0 < µ ≤ 1), and let us denote T_2,1 ≡ T₂. Furthermore, for some real p with 0< p ≤2 we define a subclass S(p) of A consisting of all function f(z) which satisfy

¯¯

¯¯ µ z

f(z)

¶₀₀¯

¯¯

¯≤p (z ∈U).

Singh [5] has shown that iff(z)∈S(p), thenf(z) satisfies (1.4)

¯¯

¯¯z²f⁰(z) (f(z))² −1

¯¯

¯¯≤p|z|², (z ∈U).

Pascu [2] has proved the following theorem:

Theorem 1.1. [2, 3] Let β ∈C, Reβ ≥γ >0. If f ∈A satisfies 1− |z|^2γ

γ

¯¯

¯¯zf⁰⁰(z) f⁰(z)

¯¯

¯¯≤1, (z ∈U), then the integral operator

F_β(z) =

· β

Z _z

0

t^β−1f⁰(t)dt

¸¹

β

is in f ∈S.

Theorem 1.2. [4] Let α, β ∈ C and Reβ ≥ Reα ≥ _|α|³ . Let f ∈ A, that satisfies the condition

¯¯

¯¯z²f⁰(z) (f(z))² −1

¯¯

¯¯<1, (z ∈U)

and |f(z)| ≤1, (z ∈U), then the integral operator

H_α,β(z) = (

β Z _z

0

t^β−1 µf(t)

t

¶¹

α

dt )¹_β

is in S.

Using Theorem1.1 and Theorem 1.2, Breaz and Breaz [1] obtained the following Theorems.

Theorem 1.3. [1] Let α, β ∈C and Reβ ≥Reα > _|α|³ⁿ. Let f_i ∈T₂ and

(1.5) f_i(z) = z+

X∞

k=3

aⁱ_kz^k

for all i= 1,2,· · · , n, n ∈N^∗ :=N\ {0} and if

|fi(z)| ≤1, (z ∈U, i= 1,2,· · · , n), then the integral operator

(1.6) Fα,β(z) = (

β Z _z

0

t^β−1 Yn

i=1

µf_i(t) t

¶¹

α

dt )¹

β

is in S.

Theorem 1.4. [1] Let α, β ∈ C and Reβ ≥ Reα > ^n(µ+2)_|α| . Let fi ∈ T2,µ

defined by (1.5) for all i = 1,2,· · · , n, n ∈ N^∗ and if |fi(z)| ≤ 1, (z ∈ U, i= 1,2,· · · , n), then the integral operator defined by (1.6) is in S.

Theorem 1.5. [1] Let α, β ∈ C and Reβ ≥ Reα > ^n(p+2)_|α| . Let f_i ∈ S(p) defined by (1.5) for all i = 1,2,· · · , n, n ∈ N^∗ and if |f_i(z)| ≤ 1, (z ∈ U, i= 1,2,· · · , n), then the integral operator defined by (1.6) is in S.

Theorem 1.2 is true even if Reβ ≥ Reα ≥ 3/|α| is replaced by the conditionReβ ≥3/|α|. Similarly Theorem1.3 is true even ifReβ≥Reα≥

3n/|α| is replaced by the condition Reβ ≥ 3n/|α|, Theorem 1.4 is true even if Reβ ≥ Reα ≥ ^n(µ+2)_|α| is replaced by the condition Reβ ≥ ^n(µ+2)_|α|

and Theorem 1.5 is true even if Reβ ≥ Reα ≥ ^n(p+2)_|α| is replaced by the condition Reβ≥ ^n(p+2)_|α| .

In this paper we extend Theorems 1.3-1.5 and also obtain the sufficient condition for univalency of certain integral operator.

To prove our main results we need the following lemma:

Lemma 1.1. (Schwarz’s Lemma) If the function w(z) is analytic in the unit desk U, w(0) = 0, and |w(z)| ≤1, for all z ∈U, then

|w(z)| ≤ |z|, (z ∈U) and equality holds only if w(z) = ²z, where |²|= 1.

2 Sufficient Conditions For Univalence

Forfi ∈A2(i= 1,2,· · · , n) andα1, α2,· · · , αn, β ∈C, we define an integral operator by

(2.1) Fα1,α2,···,αn,β(z) = (

β Z _z

0

t^β−1 Yn

i=1

µfi(t) t

¶¹

αi dt )¹

β

.

When αi = α for all i = 1,2,· · · , n, Fα1,α2,···,αn,β(z) becomes the integral operator Fα,β(z) considered in Theorem 1.3.

Theorem 2.1. Let M ≥1, f_i ∈T_2,µi defined by (1.5), α_i, β ∈C, Reβ≥γ and

(2.2) γ :=

Xn

i=1

(1 +µ_i)M + 1

|α_i| (0< µ_i ≤1, f or all i= 1,2,· · · , n, n∈N^∗).

If

|f_i(z)| ≤M, (z ∈U, i= 1,2,· · · , n),

then the integral operator F_{α1,α2,···,αn,β}(z) defined by (2.1) is in S.

Proof. Define a function h(z) =

Z _z

0

Yn

i=1

µfi(t) t

¶¹

αi dt,

then we have h(0) =h⁰(0)−1 = 0. Also a simple computation yields h⁰(z) =

Yn

i=1

µf_i(z) z

¶ ¹

αi

and

(2.3) zh⁰⁰(z)

h⁰(z) = Xn

i=1

1 αi

µzf_i⁰(z) fi(z) −1

¶ .

From equation (2.3), we have

¯¯

¯¯zh⁰⁰(z) h⁰(z)

¯¯

¯¯ ≤ Xn

i=1

1

|α_i| µ¯¯

¯¯zf_i⁰(z) f_i(z)

¯¯

¯¯+ 1

¶

= Xn

i=1

1

|α_i| µ¯¯¯

¯z²f_i⁰(z) (f_i(z))²

¯¯

¯¯

¯¯

¯¯f_i(z) z

¯¯

¯¯+ 1

¶ (2.4)

From the hypothesis, we have |f_i(z)| ≤M (z∈U, i= 1,2,· · ·, n),then by Schwarz Lemma, we obtain that

|f_i(z)| ≤M|z| (z ∈U, i= 1,2,· · · , n).

We apply this result in inequality (2.4), we obtain

¯¯

¯¯zh⁰⁰(z) h⁰(z)

¯¯

¯¯ ≤ Xn

i=1

1

|α_i| µ¯¯

¯¯z²f_i⁰(z) (f_i(z))²

¯¯

¯¯M + 1

¶

≤ Xn

i=1

1

|αi| µ¯¯

¯¯z²f_i⁰(z) (fi(z))² −1

¯¯

¯¯M +M + 1

¶ (2.5)

= Xn

i=1

1

|αi|(µ_iM +M + 1) = Xn

i=1

(1 +µ_i)M + 1

|αi| .

Because of f_i ∈T_2,µi, (1.3) in (2.5) and in view of (2.2) we have

¯¯

¯¯zh⁰⁰(z) h⁰(z)

¯¯

¯¯ <

Xn

i=1

(1 +µ_i)M + 1

|α_i| =γ.

(2.6)

Multiply (2.6) by

1− |z|^2γ

γ ,

we have

1− |z|^2γ γ

¯¯

¯¯zh⁰⁰(z) h⁰(z)

¯¯

¯¯ ≤ 1− |z|^2γ <1 (z ∈U).

Since Reβ≥γ >0 it follows from Theorem1.1 that

· β

Z _z

0

t^β−1h⁰(t)dt

¸¹

β

∈S.

Since

· β

Z _z

0

t^β−1h⁰(t)dt

¸¹

β

=

"

β Z _z

0

t^β−1 Yn

i=1

µf_i(t) t

¶¹

αi dt

#_β¹

=F_{α1,α2,···,αn,β}(z), the integral operator F_{α1,α2,···,αn,β}(z) defined by (2.1) is in S.

Remark 2.1. By taking M = 1, αi =α, f or all i= 1,2,· · · , n, then The- orem 2.1 reduces to Theorem 1.4. By taking µi =µ= 1, αi =α, f or all i= 1,2,· · · , n, then Theorem 2.1 reduces to Theorem 1.3.

Theorem 2.2. Let M ≥1, f_i ∈S(p)defined by (1.5), α_i, β ∈C, Reβ≥γ₁ and

(2.7) γ₁ :=

Xn

i=1

(1 +p)M+ 1

|αi| ( f or all i= 1,2,· · · , n, n∈N^∗).

If

|f_i(z)| ≤M (z ∈U, i= 1,2,· · · , n),

then the integral operator F_{α1,α2,···,αn,β}(z) defined by (2.1) is in S.

Proof. Define a function h(z) =

Z _z

0

Yn

i=1

µf_i(t) t

¶¹

αi dt,

then we have h(0) =h⁰(0)−1 = 0. Because of f_i ∈ S(p), (1.4) in (2.5), in view of (2.7) we have

¯¯

¯¯zh⁰⁰(z) h⁰(z)

¯¯

¯¯ ≤ Xn

i=1

1 +M +Mp|z|²

|α_i| (2.8)

<

Xn

i=1

(1 +p)M+ 1

|α_i| =γ₁ (z ∈U).

(2.9)

Rest of the proof is similar to Theorem 2.1, then we omit the details.

Remark 2.2. By taking M = 1, α_i = α, f or all i = 1,2,· · · , n, then Theorem 2.2 reduces to Theorem 1.5.

Theorem 2.3. Let α_i, β ∈C, Reβ ≥γ₂ and (2.10) γ₂ :=

Xn

i=1

β_i

|α_i| (0< β_i ≤1, f or all i= 1,2,· · · , n, n∈N^∗).

If f_i ∈A₂ defined by (1.5) satisfy the conditions (2.11)

¯¯

¯¯zf_i⁰(z) f_i(z) −1

¯¯

¯¯≤βi (0< βi ≤1, z ∈U, i= 1,2,· · · , n), then the integral operator F_{α1,α2,···,αn,β}(z) defined by (2.1) is inS.

Proof. From (2.3), we get (2.12)

¯¯

¯¯zh⁰⁰(z) h⁰(z)

¯¯

¯¯=

¯¯

¯¯

¯ Xn

i=1

1 α_i

µzf_i⁰(z) f_i(z) −1

¶¯¯¯

¯¯≤ Xn

i=1

1

|α_i|

¯¯

¯¯zf_i⁰(z) f_i(z) −1

¯¯

¯¯. Substituting (2.11) in (2.12) and in view of (2.10) we have

¯¯

¯¯zh⁰⁰(z) h⁰(z)

¯¯

¯¯ <

Xn

i=1

β_i

|αi| =γ₂. (2.13)

Rest of the proof is similar to Theorem 2.1, then we omit the details.

By taking β_i = 1 and α_i =α (f or all i = 1,2,· · · , n) in Theorem 2.3, we obtained the following result.

Example 2.1. Let α, β ∈C, Reβ≥ _|α|ⁿ. If f_i ∈A₂ defined by (1.5) satisfy the conditions

(2.14)

¯¯

¯¯zf_i⁰(z) fi(z) −1

¯¯

¯¯≤1 (z ∈U, i= 1,2,· · · , n), then the integral operator F_α,β(z) defined by (1.6) is in S.

References

[1] D. Breaz and N. Breaz, The univalent conditions for an integral oper- ator on the classes S(p) and T₂, Journal of Approximation Theory and Applications, Vol. 1, No.2, (2005), pp. 93–98.

[2] N. N. Pascu,On a univalence criterion. II, in Itinerant seminar on func- tional equations, approximation and convexity (Cluj-Napoca, 1985), 153–154, Univ. “Babe¸s-Bolyai”, Cluj.

[3] N. N. Pascu, An improvement of Becker’s univalence criterion, in Proceedings of the Commemorative Session: Simion Sto¨ılow (Bra¸sov, 1987), 43–48, Univ. Bra¸sov, Bra¸sov.

[4] V. Pescar, New criteria for univalence of certain integral operators, Demonstratio Math. 33(1) (2000), 51–54.

[5] V. Singh, On a class of univalent functions, Int. J. Math. Math. Sci.

23(12) (2000), 855–857.

[6] S. Ozaki and M. Nunokawa, The Schwarzian derivative and univalent functions, Proc. Amer. Math. Soc. 33 (1972), 392–394.

Narayanasamy Seenivasagan 2A Kamarajar Street

Samatharmapuram, Theni 625531 India

E-mail: [email protected]

Daniel Breaz

Department of Mathematics and Computer Science

”1 Decembrie 1918” University of Alba Iulia 510009, str. N. Iorga, No. 11-13, Alba, Romania E-mail: [email protected]