© 2015. Dr. Poonam Dixit & Puneet Shukla. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 15 Issue 8 Version 1.0 Year 2015 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896

The Univalence of A Generalized Integral Operator

By Dr. Poonam Dixit & Puneet Shukla C.S.J.M. University, India

Abstract- For analytic function , in the open disk , an integral operator is introduced. In this paper we obtain the conditions of the univalence for

the integral operator .

Keywords: fuzzy anti 2-banach space.

GJSFR-F Classification : MSC 2010: 11S23 TheUnivalenceofAGeneralizedIntegralOperator

Strictly as per the compliance and regulations of :

fi, j= 1, n U Kα1, .....αn ,β1.....βn

Kα1 , .....αn ,β1 .....βn

The Univalence of a Generalized Integral Operator Dr. Poonam Dixit

α & Puneet Shukla

σ

Abstract- For analytic function in the open disk , an integral operator is introduced. In this paper we obtain the conditions of the univalence for the integral operator . Keywords: fuzzy anti 2-banach space.

I. Introduction

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Author α σ: Department of Mathematics UIET, C.S.J.M. University, Kanpur. e-mails: dixit_poonam14@rediffmail.com, puneetshukla05@gmail.com

fi, j= 1, n U Kα1, .....αn ,β1.....βn

Kα1 , .....αn ,β1 .....βn

Let A be the class of functions f of the form

f(z) = z +∞∑

k=2

akzk

which are analytic in the open disk U = {z ∈ C : |z| < 1} with f(0) = f′(0)− 1 = 0. Let S

denote the subclass of A consisting of the functions f ∈ A , which are univalent in U .Wedenote by P the class of functions p which are analytic in U , p(0) = 1 and Rep(z) > 0 , forall z ∈ U . In this work, we introduce a new integral operator, which is given by

Kα1 , .....αn ,β1 .....βn(z) =

∫ z

0

n∏j=1

(Dmfi(u)

u

)αj [(Dnfj(u))

′]βj

du (1)

for αj, βj be complex numbers, fi ∈ A, f′j ∈ P, j = 1, n.

For m = 1 βj = 0 j = 1, n we obtain the integral operator, which is de�ned in [4].

For m = 1 αj = 0 j = 1, n we have the integral operator, which is de�ned in [5].

II. Preliminary Results

In order to prove our main results we will need the following lemmas .[1] If the function f is analytic in U and

(1− |z|)2

∣∣∣∣zf ′′(z)

f ′(z)− 1

∣∣∣∣ ≤ 1 (2)

Lemma 2.1

4.V

. P

esca

r an

d V

. D

. B

reaz

. T

he

univ

alen

ce of

in

tegr

al op

erat

ors,

M

onog

raph,

Aca

dem

icP

ublish

ing

Hou

se, Sof

ia 2

008.

for all z ∈ U , then the function f is univalent in U .[3] Let γ be a complex number Reγ > 0 and f ∈ A . If

1− |z|2Re γ

Re γ

∣∣∣∣zf ′′(z)

f ′(z)

∣∣∣∣ ≤ 1 (3)

for all z ∈ U , then for any complex number δ, Re δ ≥ Re γ , the function

fδ(z) =

[δ

∫ z

0

uδ−1f′(u)du

]1/δ

(4)

is regular and univalent in U .

(Schwarz [2]) Let f be the function regular in the disk UR = {z ∈ C : |z| < R}with |f(z)| < M,M �xed .

If f has in z = 0 one zero multiply ≥ m then

|f(z)| ≤ M

Rm|z|m (z ∈ UR) (5)

the equality (in the inequality (5)z 6= 0) can hold only if .

f(z) = eiθ MRm zm

where θ is constant.

Let αj, βj be the complex numbers Mj, Lj positive real numbers, j = 1, n

and the functions

fj ∈ A, f′j ∈ P, fj(z) = z + a2jz

2 + a3jz3 + ....j = 1, n if .∣∣∣∣zDm+1fj(z)

Dmfj(z)− 1

∣∣∣∣ ≤ Mj (j = 1, n : z ∈ U) (6)

The Univalence of a Generalized Integral Operator

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Lemma 2.2

Lemma 2.3

III. Main Results

∣∣∣∣z [Dnfj(z)]′′

[Dnfj(z)]′

∣∣∣∣ ≤ Lj (j = 1, n : z ∈ U) (7)

andn∑

j=1

[|αj|Mj + |βj|Lj] ≤3√

3

2(8)

Then the integral operator Kα1 , .....αn ,β1 .....βn de�ned by (1) is in the class S .The function Kα1 , .....αn ,β1 .....βn(z) is regular in U and

Kα1 , .....αn ,β1 .....βn(0) = K′α1

, .....αn ,β1 .....βn(0)− 1 = 0

Theorem 3.1 :

Proof :

2.O

. May

er, The fu

nction

s theory

of one v

ariable com

plex

, Bucu

resti. 1981.

Ref

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we have

zK′′α1

, .....αn ,β1 .....βn(z)

K ′α1

, .....αn ,β1 .....βn(z)=

n∑j=1

[αj

(Dm+1fj(z)

Dmfj(z)− 1

)]+

n∑j=1

[βjz

[Dnfj(z)]′′

[Dnfj(z)]′

](9)

and hence we get

(1−|z|2)

∣∣∣∣∣zK′′α1

, .....αn ,β1 .....βn(z)

K ′α1

, .....αn ,β1 .....βn(z)

∣∣∣∣∣ ≤ (1−|z|2)n∑

j=1

[|αj|

∣∣∣∣zDm+1fj(z)

Dmfj(z)− 1

∣∣∣∣ + |βj|∣∣∣∣z [Dnfj(z)]

′′

[Dnfj(z)]′

∣∣∣∣](10)

for all z ∈ U.

By (6), (7) and Lemma 2.3, we obtain

∣∣∣∣zDm+1fj(z)

Dmfj(z)− 1

∣∣∣∣ ≤ Mj |z| (j = 1, n : z ∈ U) (11)

∣∣∣∣z [Dnfj(z)]′′

[Dnfj(z)]′

∣∣∣∣ ≤ Lj |z| (j = 1, n : z ∈ U) (12)

and from (10) we have

(1− |z|2)

∣∣∣∣∣zK′′α1

, .....αn ,β1 .....βn(z)

K ′α1

, .....αn ,β1 .....βn(z)

∣∣∣∣∣ ≤ (1− |z|2) |z|

{n∑

j=1

[|αj|Mj + |βj|Lj]

}(13)

for all z ∈ U .

Since

max|z|<1[(1− |z|2) |z|] =2

3√

3

from (8) and (13) we get

(1− |z|2)

∣∣∣∣∣zK′′α1

, .....αn ,β1 .....βn(z)

K ′α1

, .....αn ,β1 .....βn(z)

∣∣∣∣∣ ≤ 1, (z ∈ U)

and by Lemma 2.1, it results that the integral operator Kα1 , .....αn ,β1 .....βn is in the class S.

Let αj, βj, γ be the complex numbers j = 1, n , 0 < Re γ ≤ 1 and thefunctions

fj ∈ A, f′j ∈ P, fj(z) = z + a2jz

2 + a3jz3 + ....j = 1, n

if ∣∣∣∣zDm+1fj(z)

Dmfj(z)− 1

∣∣∣∣ ≤ (2Re γ + 1)2Re γ+12Re γ

2(j = 1, n : z ∈ U) (14)

Theorem 3.2 :

Notes

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∣∣∣∣z [Dnfj(z)]′′

[Dnfj(z)]′

∣∣∣∣ ≤ (2Re γ + 1)2Re γ+12Re γ

2(j = 1, n : z ∈ U) (15)

andn∑

j=1

[|αj|+ |βj|] ≤ 1 (16)

then the integral operatorKα1 , .....αn ,β1 .....βn de�ned by (1) belong to the class S .

From (9) we obtain

zK′′α1

, .....αn ,β1 .....βn(z)

K ′α1

, .....αn ,β1 .....βn(z)=

n∑j=1

[αj

(zDm+1fj(z)

Dmfj(z)− 1

)]+

n∑j=1

[βj

z[Dnfj(z)]′′

[Dnfj(z)]′

]and hence we get

1− |z|2Reγ

Re γ

∣∣∣∣∣zK′′α1

, .....αn ,β1 .....βn(z)

K ′α1

, .....αn ,β1 .....βn(z)

∣∣∣∣∣≤1− |z|2Reγ

Re γ

n∑j=1

[|αj|

∣∣∣∣zDm+1fj(z)

Dmfj(z)−1

∣∣∣∣+|βj|∣∣∣∣z [Dnfj(z)]

′′

[Dnfj(z)]′

∣∣∣∣](17)

for all z ∈ U by (14), (15) and Lemma 2.3 we have

∣∣∣∣zDm+1fj(z)

Dmfj(z)− 1

∣∣∣∣ ≤ (2Re γ + 1)2Reγ+12Re γ

2|z| (j = 1, n : z ∈ U) (18)

∣∣∣∣z [Dnfj(z)]′′

[Dnfj(z)]′

∣∣∣∣ ≤ (2Re γ + 1)2Reγ+12Re γ

2|z| (j = 1, n : z ∈ U) (19)

and hence by (17) we get

1− |z|2Reγ

Re γ

∣∣∣∣∣zK′′α1

, .....αn ,β1 .....βn(z)

K ′α1

, .....αn ,β1 .....βn(z)

∣∣∣∣∣ ≤ 1− |z|2Reγ

Re γ|z| (2Re γ + 1)

2Reγ+12Re γ

2

n∑j=1

[|αj|+|βj|] (20)

for all z ∈ U .

max|z|≤1

[1− |z|2Reγ

Re γ|z|

]=

2

(2Re γ + 1)2Reγ+12Re γ

From (16) and (20) we obtain that

1− |z|2Reγ

Re γ

∣∣∣∣∣zK′′α1

, .....αn ,β1 .....βn(z)

K ′α1

, .....αn ,β1 .....βn(z)

∣∣∣∣∣ ≤ 1 (21)

Proof :

Notes

The Univalence of a Generalized Integral Operator

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for all z ∈ U and by Lemma 2.2 for δ = 1 and f = Kα1 , .....αn ,β1 .....βn it results that theintegral operator Kα1 , .....αn ,β1 .....βn de�ned by (1) belongs to the class S .

Let αj be the complex numbers Mj positive real numbers, j = 1, n andthe functions

fj ∈ A, f′j ∈ P, fj(z) = z + a2jz

2 + a3jz3 + ....j = 1, n if .

∣∣∣∣zDm+1fj(z)

Dmfj(z)− 1

∣∣∣∣ ≤ Mj (j = 1, n : z ∈ U) (22)

and

n∑j=1

[|αj|Mj] ≤3√

3

2(23)

then the function

Gα1 , .....αn(z) =

∫ z

0

n∏j=1

(Dmfi(u)

u

)αj

du

is in the class S.

Let βj be the complex numbers Lj positive real numbers, j = 1, n and thefunctions

fj ∈ A, f′j ∈ P, fj(z) = z + a2jz

2 + a3jz3 + ....j = 1, n

and ∣∣∣∣z [Dnfj(z)]′′

[Dnfj(z)]′

∣∣∣∣ ≤ Lj (j = 1, n : z ∈ U) (24)

andn∑

j=1

[|βj|Lj] ≤3√

3

2(25)

then the function

Hβ1 , .....βn(z) =

∫ z

0

n∏j=1

[(Dnfi(u))

′]βj

du

belongs to the class S.

IV. Corollaries

Corollary 4.1 :

Corollary 4.2 :

Notes

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References Références Referencias

1. J. Becker, L Lownersche differentialgleichung and quasikonforrn fortsetzbare schlichte functionen, J. Reine Angew. Math., 255, 1972,23-43.

2. O. Mayer, The functions theory of one variable complex, Bucuresti. 1981.3. N. N. Pascu, An improvement of Becker's univalence criterion, proceedings of the

com-memorative session stoilow, University of Brasov. 1987, 43-48. 4. V. Pescar and V. D. Breaz. The univalence of integral operators, Monograph,

Academic Publishing House, Sofia 2008.5. V. Pescar, On the univalence of an integral operator, General Math. 19 (4) (2011),

69-74.

Notes

The Univalence of a Generalized Integral Operator