Univalence criteria for a new integral operator

Mathematical and Computer Modelling 52 (2010) 241–246

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Univalence criteria for a new integral operator Nicoleta Breaz a , Virgil Pescar b , Daniel Breaz a,∗ a

Department of Mathematics, ‘‘1 Decembrie 1918’’ University of Alba Iulia, 510009, Alba Iulia, Romania

b

Department of Mathematics, ‘‘Transilvania’’ University of Braşov, 500091 Braşov, Romania

article

abstract

info

Article history: Received 18 November 2009 Received in revised form 5 February 2010 Accepted 9 February 2010

In view of an integral operator Hγ1 ,γ2 ,...,γ|[Re η]| ,β,η for an analytic function f in the open unit disk U, sufficient conditions for univalence of this integral operator are discussed. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Integral operator Univalence Starlike Real part Integer part

1. Introduction Let A be the class of functions f of the form f (z ) = z +

∞ X

an z n

n =2

which are analytic in the open unit disk U = {z ∈ C : |z | < 1}. Let S denote the subclass of A consisting of all univalent functions f in U. For f ∈ A, the integral operator Gα is defined by Gα (z ) =

Z z 0

f (u) u

 α1 du

(1.1)

for some complex numbers α (α 6= 0). 1 ≤ In [1] Kim–Merkes have proved that the integral operator Gα is in the class S for |α| In [2] Pascu and Pescar defined the integral operator

1 4

and f ∈ S .

 Z z α  β1  β−1 f (u) Hα,β (z ) = β u du 0

u

for α, β being complex numbers, β 6= 0 and f ∈ A. In [2–5] we have the sufficient conditions of univalence for the integral operator Hα,β .

∗

Corresponding author. E-mail addresses: [email protected] (N. Breaz), [email protected] (V. Pescar), [email protected], [email protected] (D. Breaz).

0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.02.013

(1.2)

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N. Breaz et al. / Mathematical and Computer Modelling 52 (2010) 241–246

Also, the integral operator Jγ for f ∈ A is given by Mγ (z ) =



1

z

Z

γ

1

u−1 (f (u)) γ du

γ (1.3)

0

where γ is a complex number, γ 6= 0. Miller and Mocanu [6] have shown that the integral operator Mγ is in the class S for f ∈ S ∗ , γ > 0, S ∗ is the subclass of S consisting of all starlike functions f in U. We introduce the general integral operator

( Hγ1 ,γ2 ...,γ|[Re η]| ,β,η (z ) =

ηβ

z

Z

u

ηβ−1



f1 (u)

 γ1

1

u

0

...



f|[Re η]| (u)

γ

) ηβ1

1

|[Re η]|

u

du

(1.4)

for fj ∈ A, γj , η, β complex numbers, where γj 6= 0, Re η 6∈ [0, 1), j = 1, |[Re η]|, β 6= 0, |[Re η]| is the modulus of integer part of η. If in (1.4) we take η = 1, γ1 = α1 , f1 = f we obtain the integral operator Hα,β . For ηβ = 1, |[Re η]| = 1, γ1 = α , f1 = f , for (1.4), we have the integral operator Gα . For ηβ = γ1 , γ1 = γ , |[Re η]| = 1, f1 = f , from (1.4) we obtain the integral operator Mγ . 2. Preliminary results We need the following lemmas. Lemma 2.1 ([7]). Let α be a complex number, Re α > 0 and f ∈ A. If 1 − |z |2Re α zf 00 (z )

f 0 (z ) ≤ 1

Re α

(2.1)

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

 Z

Fβ (z ) = β

uβ−1 f 0 (u)du

 β1 (2.2)

0

is in the class S. Lemma 2.2 (Schwarz [8]). Let f be a function regular in the disk UR = {z ∈ C : |z | < R} with |f (z )| < M, M fixed. If f (z ) has one zero with multiplicity order bigger than m, for z = 0, then

|f (z )| ≤

M Rm

|z |m ,

z ∈ UR

(2.3)

the equality (in the inequality (2.3) for z 6= 0) can hold only if f (z ) = eiθ

M Rm

zm,

where θ is constant. 3. Main results Theorem 3.1. Let γj , β, η be complex numbers, β 6= 0, Re η 6∈ [0, 1), j = 1, |[Re η]|, a =

P|[Re η]| j =1

Re γ1 > 0 and j

fj ∈ A, fj (z ) = z + b2j z 2 + b3j z 3 + · · · , j = 1, |[Re η]|. If

0 2a+1 zfj (z ) (2a + 1) 2a ≤ γj , − 1 f (z ) 2 |[Re η]| j

j = 1, |[Re η]|,

(3.1)

for all z ∈ U, and Re ηβ ≥ a, then the function

( Hγ1 ,γ2 ,...,γ|[Re η]| ,β,η (z ) = is in the class S .

ηβ

z

Z 0

uηβ−1



f1 (u) u

 γ1

1

...



f|[Re η]| (u) u

γ

) ηβ1

1

|[Re η]|

du

(3.2)

N. Breaz et al. / Mathematical and Computer Modelling 52 (2010) 241–246

243

Proof. We consider the function g (z ) =

Z z

f 1 ( u)

 γ1

1

...

u

0



f|[Re η]| (u)

γ

1

|[Re η]|

u

du.

The function g is regular in U. We define the function p(z ) = p(z ) =

zg 00 (z ) g 0 (z )

=

|[Re Xη]| 

1



γj

j =1

zfj0 (z ) fj (z )

 −1

,

(3.3) zg 00 (z ) g 0 (z )

, z ∈ U and we obtain

z ∈ U.

(3.4)

From (3.1) and (3.4) we have

|p(z )| ≤

(2a + 1)

2a+1 2a

(3.5)

2

for all z ∈ U and applying Lemma 2.2 we get

|p(z )| ≤

(2a + 1)

2a+1 2a

|z |, 2 From (3.4) and (3.6) we have

z ∈ U.

(3.6)

2a+1 2a 2a ≤ (1 − |z | )|z | · (2a + 1) g 0 (z ) a 2

1 − |z |2a zg 00 (z ) a

(3.7)

for all z ∈ U. Because max

(1 − |z |2a )|z |

|z |≤1

a

=

2

(2a + 1)

2a+1 2a

,

from (3.7) we have 1 − |z |2a zg 00 (z ) a

g 0 (z ) ≤ 1

(3.8)

for all z ∈ U. So, by the Lemma 2.1, the integral operator Hγ1 ,γ2 ,...,γ|[Re η]| ,β,η belongs to the class S .



Corollary 3.2. Let γ be a complex number, a = Re γ1 > 0 and f ∈ A, f (z ) = z + b21 z 2 + b31 z 3 + · · ·. If

0 2a+1 (2a + 1) 2a zf (z ) |γ | f (z ) − 1 ≤ 2

(3.9)

for all z ∈ U, then the integral operator Mγ defined by (1.3) belongs to the class S . Proof. We take η = 1, β = γ1 , γ1 = γ , f1 = f in Theorem 3.1.



Corollary 3.3. Let α, β be complex numbers, β 6= 0, a = Re α > 0, f ∈ A, f (z ) = z + b21 z 2 + b31 z 3 + · · ·. If

0 2a+1 zf (z ) (2a + 1) 2a ≤ − 1 f (z ) 2 |α|

(3.10)

for all z ∈ U, then for Re β ≥ Re α the integral operator Hα,β is in the class S . Proof. For η = 1, γ1 = α1 , f1 = f , from Theorem 3.1 we have Corollary 3.3.



Corollary 3.4. Let α be a complex number, a = Re α1 ∈ (0, 1], f ∈ A, f (z ) = z + b21 z 2 + b31 z 3 + · · ·. If

0 2a+1 zf (z ) (2a + 1) 2a ≤ |α| − 1 f (z ) 2 for all z ∈ U, then the integral operator Gα given by (1.1) is in the class S .

(3.11)

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N. Breaz et al. / Mathematical and Computer Modelling 52 (2010) 241–246

Proof. We take ηβ = 1, |[Re η]| = 1, γ1 = α , f1 = f in Theorem 3.1.



Corollary 3.5. Let α, η be complex numbers a = |[Re η]|· Re α1 , Re η 6∈ [0, 1), a ∈ (0, 1] and fj ∈ A, fj (z ) = z + b2j z 2 +· · · , j = 1, |[Re η]|. If

0 2a+1 zfj (z ) (2a + 1) 2a ≤ |α|, − 1 f (z ) 2 |[Re η]| j

j = 1, |[Re η]|

(3.12)

for all z ∈ U, then the function Lα ( z ) =

Z z

f 1 ( u)

 α1

u

0

...



f|[Re η]| (u)

 α1 du

u

(3.13)

is in the class S . Proof. For ηβ = 1, γ1 = γ2 = · · · = γ|[Re η]| = α from Theorem 3.1. we obtain the Corollary 3.5.



Theorem 3.6. Let γj , α, β, η be complex numbers, γj 6= 0, Re η 6∈ [0, 1), β 6= 0, a = Re α > 0, j = 1, |[Re η]| and P∞ fj ∈ S , fj (z ) = z + k=2 bkj z k , j = 1, |[Re η]|. If |[Re Xη]|

1

j =1

|γj |

|[Re Xη]|

1

j =1

|γj |

≤

a 2

,

for 0 < a <

,

for a ≥

1

(3.14)

2

or

≤

1 4

1

(3.15)

2

then for Re ηβ ≥ a, the integral operator Hγ1 ,γ2 ,...,γ|[Re η]| ,β,η given by (1.4) is in the class S . Proof. We consider the function g (z ) =

Z z

f1 (u)

 γ1

u

0

1

...



f|[Re η]| (u)

γ

1

|[Re η]|

u

du.

(3.16)

The function g is regular in U. We have

 2a |[Re Xη]|  1 zfj0 (z ) ≤ 1 − |z | . − 1 g 0 (z ) a |γ | f ( z ) j j j =1

1 − |z |2a zg 00 (z ) a

(3.17)

Since fj ∈ S , j = 1, |[Re η]| we have

0 zfj (z ) 1 + |z | f (z ) ≤ 1 − |z | , j

z ∈ U, j = 1, |[Re η]|.

(3.18)

From (3.17) and (3.18) we obtain

|[Re 2a Xη]| 1 2 1 − |z | g 0 (z ) ≤ a 1 − |z | j=1 |γj |

1 − |z |2a zg 00 (z ) a for all z ∈ U. For 0 < a < max |z |≤1

1 2

(3.19)

we have

1 − |z | 2a 1 − |z |

=1

and from (3.14), (3.19) we get 1 − |z |2a zg 00 (z ) a

g 0 (z ) ≤ 1,

z ∈ U.

(3.20)

N. Breaz et al. / Mathematical and Computer Modelling 52 (2010) 241–246

For a ≥ max |z |≤1

1 2

245

we have

1 − |z | 2a 1 − |z |

= 2a

and from (3.15), (3.19) we obtain (3.20). From (3.20) and Lemma 2.1 it results that the integral operator Hγ1 ,γ2 ,...,γ|[Re η]| ,β,η belongs to the class S . Corollary 3.7. Let α, γ be complex numbers, a = Re α > 0 and f ∈ S , f (z ) = z + If 1

|γ |

≤

a 2

,

for 0 < a <

,

for a ≥

P∞

k =2



bk1 z k .

1

(3.21)

2

or 1

|γ |

≤

1 4

1

(3.22)

2

then for Re γ1 ≥ a, the integral operator Mγ given by (1.3) is in the class S . Proof. We take in Theorem 3.6, ηβ = γ1 , |[Re η]| = 1, γ1 = γ , f1 = f .



Corollary 3.8. Let α, β, γ be complex numbers, a = Re γ > 0 and f ∈ S , f (z ) = z + If

|α| ≤

a 2

,

for 0 < a <

,

for a ≥

P∞

k=2

bk1 z k .

1

(3.23)

2

or

|α| ≤

1 4

1

(3.24)

2

then for Re β ≥ a, the integral operator Hα,β given by (1.2) belongs to the class S . Proof. For η = 1, f1 = f , γ1 = α1 , from Theorem 3.6 we obtain Corollary 3.8.



Corollary 3.9. Let α, γ be complex numbers, a = Re γ ∈ (0, 1] and f ∈ S , f (z ) = z + If 1

|α|

≤

a 2

,

for 0 < a <

,

for a ≥

P∞

k=2

bk1 z k .

1

(3.25)

2

or 1

|α|

≤

1 4

1

(3.26)

2

then the integral operator Gα given by (1.1) is in the class S . Proof. For ηβ = 1, |[Re η]| = 1, γ1 = α , f1 = f , from Theorem 3.6 we obtain Corollary 3.9.



Corollary 3.10. Let α, η, γ be complex numbers, Re η 6∈ [0, 1), a = Re γ ∈ (0, 1], fj ∈ S , fj (z ) = z + j = 1, |[Re η]|. If

|[Re η]| a ≤ , |α| 2

for 0 < a <

|[Re η]| 1 ≤ , |α| 4

for a ≥

P∞

k =2

1

bkj z k ,

(3.27)

2

or 1

(3.28)

2

then the integral operator Lα given by (3.13) is in the class S . Proof. For ηβ = 1, Re η 6∈ [0, 1), γ1 = γ2 = · · · = γ|[Re η]| = α , from Theorem 3.6. we obtain the Corollary 3.10.



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References [1] Y.J. Kim, E.P. Merkes, On an integral of powers of a spirallike function, Kyungpook Mathematical Journal 12 (1972) 249–253. [2] N.N. Pascu, V. Pescar, On the integral operators of Kim–Merkes and Pfaltzgraff, 32 (55), 2, Mathematica, Univ. Babeş-Bolyai, Cluj-Napoca, 1990, pp. 185–192. [3] V. Pescar, Univalence conditions for certain integral operators, Journal of Inequalities in Pure and Applied Mathematics 7 (4) (2006) Article 147. [4] V. Pescar, Integral operators on a certain class of analytic functions, Libertas Mathematica XXVII (2007) 95–98. [5] V. Pescar, S. Owa, Univalence problems for integral operators by Kim–Merkes and Pfaltzgraff, Journal of Approximation Theory and Applications 3 (1–2) (2007) 17–21. [6] S.S. Miller, P.T. Mocanu, Differential Subordinations, Theory and Applications, in: Monographs and Text Books in Pure and Applied Mathematics, vol. 225, Marcel Dekker, New York, 2000. [7] N.N. Pascu, An improvement of Becker’s univalence criterion, in: Proceedings of the Commemorative Session Simion Stoilow, University of Braşov, 1987, pp. 43–48. [8] O. Mayer, The Functions Theory of One Variable Complex, Bucureşti, 1981.