Some Properties for Certain General Integral Operator

Vol. 6, No. 3, 2013, 307-314

ISSN 1307-5543 – www.ejpam.com

Some Properties for Certain General Integral Operator

Vasile Marius Macarie

_{, Daniel Breaz}

1_{Department of Mathematics and Computer Sciences, University of Pite¸sti, Pite¸sti, România} 2_{Department of Mathematics, "1 Decembrie 1918" University of Alba Iulia, Alba Iulia, România}

Abstract. In this paper we consider some subclasses of the class of analytic functions defined in the open unit disk of the complex plane and we study some properties for an integral operator on these classes. Particular results are presented.

2010 Mathematics Subject Classifications: 30C45

Key Words and Phrases: integral operator, analytic function, convex function, starlike function

1. Introduction

LetA denote the class of the functions f of the form

f(z) =z+ ∞

X

n=2 anzn

which are analytic in the open unit disk U = _{z _∈ C : |z| < 1}. We also denote by S the subclass of_A consisting of functions which are univalent inU.

A function f ∈ A is said to beconvex of orderα, 0≤α <1 if it satisfies the condition

Re

‚

z f00(z) f0(z) +1

Œ

> α, (z∈U)

and we denote this class byK(α).

A function f ∈ A is said to bestarlike of orderα, 0≤α≤1 if it satisfies the condition

Re

‚

z f0(z) f(z)

Œ

> α, (z_∈U)

and denote this class byS∗(α). ∗_{Corresponding author.}

Email addresses:[email protected](V. Macarie),[email protected](D. Breaz)

Let_N(ρ)be the subclass of_A consisting of the functions f which satisfy the inequality Re

‚

1+ z f

00_(z)

f0(z)

Œ

< ρ, ρ >1, (z∈U).

This class was studied by S. Owa and H.M. Srivastava in[5].

A. Mohammed et al. considered in [4] _{M T}(µ,β) the subclass of _A consisting of the functions f which satisfy the inequality

z f0(z) f(z) −1

< β

µ

z f0(z) f(z) +1

, 0< β_≤1, 0_≤µ <1, (z_∈U).

Also, Frasin and Jahangiri introduced in [2] the family _B(µ,α), µ _≥ 0, 0 _≤ α _≤ 1, consisting of the functions f which satisfy the condition

f0(z)

_z

f(z)

µ

−1

<

1−α, (z∈U).

This family is a comprehensive class of analytic functions that includes various classes of analytic functions. We have_B(1,α)_≡S∗(α)and_B(0,α)_≡R(α).

Letβ₋Sp(α)be the subclass ofA consisting of the functionsf which satisfy the inequality

Re

‚

z f0(z) f(z) −α

Œ

≥β

z f0(z) f(z) −1

, −1≤α≤1,β >0, (z∈U). This class was studied by M. Darus in[1].

A function f is said to be in the classK D(µ,α)if it satisfies the inequality Re

‚

z f00(z) f0(z) +1

Œ

≥µ

z f00(z) f0(z)

+α

, µ_≥0, 0_≤α <1, (z∈U). This class was studied by S. Shams et al. in[6].

In the present paper we study some properties for the integral operatorG_n defined by

Gn(z) =

Z z

0 n

Y

i=1 fi(t)

γi−1(_g0

i(t))ηidt (1)

on the classes presented above.

In order to prove our main results we need the following lemma:

Lemma 1(General Schwarz Lemma[3]). Let the function f be regular in the disk

U_R=_{z_∈C:|z|<R}, with|f(z)|< M for fixed M . If f has one zero with multiplicity order

bigger than m for z=0, then

|f(z)| ≤ M

Rm · |z| m _(z

∈UR). The equality can hold only if

f(z) =eiθ _· M Rm·z

2. Main Results

Theorem 1. Letγ_i∈R,γi>1,ηi∈R,ηi>0for all i=1, 2, . . . ,n, the functions

fi ∈ M T(µi,βi), 0< βi ≤ 1, 0 ≤ µi < 1and gi ∈ A for all i = 1, 2, . . . ,n satisfying the

conditions

f_i0(z) f_i(z)

<M_i, (M_i≥1)for all i=1, 2, . . . ,n (2)

and

g_i00(z)

g_i0(z)

<N_i, (N_i≥1)for all i=1, 2, . . . ,n. (3)

Then the integral operator G_n defined in(1)is in_N(ρ), where

ρ=1+

n

X

i=1

(γi−1)(βiµiMi+βi+1) +ηiNi

Proof. From (1), we have

G_n0(z) = n

Y

i=1

f_i(z)γi−1(_g0 i(z))ηi

and

zG_n00(z) G_n0(z) =

n

X

i=1

(γi−1) z f_i0(z)

f_i(z) + n

X

i=1 ηi

z g00_i (z) g0_i(z) .

Thus, we have

Re

‚

zG00_n(z) G0_n(z) +1

Œ

=

n

X

i=1

(γi−1)Re

‚

z f_i0(z)

f_i(z)

Œ

+

n

X

i=1 ηiRe

‚

z g00_i (z)

g0_i(z)

Œ

+1.

Since Rew≤ |w|, then

Re

‚

zG_n00(z)

G_n0(z) +1

Œ

≤

n

X

i=1

(γi−1)

z f_i0(z)

fi(z)

+ n X

i=1 ηi

z g_i00(z)

g_i0(z)

+1. (4)

Using that fi∈ M T(µi,βi)for alli=1, 2, . . . ,nin relation (4), we obtain

Re

‚

zG00_n(z)

G0_n(z) +1

Œ

≤

n

X

i=1

(γi−1)

z f_i0(z)

f_i(z) −1

+1 ! + n X

i=1 ηi

z g00_i (z)

g0_i(z)

+1 < n X

i=1

(γi−1)βi

µi z f_i0(z)

f_i(z) +1

+ n X

i=1

(γi−1)

+

n

X

i=1 ηi

z g_i00(z)

g0_i(z)

and using the hypothesis (2) and (3) in this last relation we have

Re

‚

zG_n00(z) G0

n(z)

+1

Œ

<

n

X

i=1

(γi−1)(βiµiMi+βi+1) +ηiNi

+1=ρ

This completes the proof of our theorem.

Letting n=1,γ₁ =γ,η₁=η, M₁ = M, N₁ =N,µ₁ =µ, β₁ =β, f₁ = f and g₁ = gin Theorem 1, we have

Corollary 1. Let γ _∈ R, γ > 1, η ∈ R, η > 0, the functions f ∈ M T(µ,β), 0 < β ≤ 1, 0_≤µ <1and g ∈ A satisfying the conditions

f0(z) f(z)

<

M, (M ≥1)and

g00(z) g0(z)

<

N, (N≥1).

Then the integral operator

G₁(z) =

Z z

0

f(t)γ−1

(g0(t))ηdt

is inN(ρ), whereρ= (γ−1)(βµM+β+1) +ηN+1.

Lettingγ=2,η=1 in Corollary 1, we have

Corollary 2. Let f ∈ M T(µ,β),0< β_≤1,0_≤µ <1and g∈ A satisfying the conditions

f0(z) f(z)

<

M, (M ≥1)and

g00(z) g0(z)

<

N, (N≥1).

Then the integral operator

G(z) =

Z z

0

f(t)g0(t)dt is inN(ρ), whereρ=β(µM+1) +N+2.

Theorem 2. Letγi∈R,γi>1,ηi∈R,ηi>0for all i=1, 2, . . . ,n, the functions

f_i ∈ B(µi,αi), µi ≥ 0, 0 ≤ αi < 1 satisfying the conditions |fi(z)| ≤ Mi, (Mi ≥ 1) and gi ∈ N(ρi),ρi>1for all i=1, 2, . . . ,n. If

n

X

i=1

h

(γi−1)(2−αi)Mµi− 1

i +ηi(ρi−1)

i

<1,

then the integral operator Gndefined in(1)is in K(δ), where

δ=1₋

n

X

i=1

h

(γi−1)(2−αi)Mµi− 1

i +ηi(ρi−1)

i

Proof. From (1), we have

G_n0(z) = n

Y

i=1

f_i(z)γi−1(_g0 i(z))ηi

and

zG_n00(z)

G_n0(z) = n

X

i=1

(γi−1) z f_i0(z)

f_i(z) + n

X

i=1 ηi

z g00_i (z)

g0_i(z) .

Hence

zG_n00(z)

G_n0(z)

≤

n

X

i=1

(γi−1)

‚

f_i0(z)

_z

fi(z)

µi

−1

+

1

Œ

fi(z) z

µi−1

+

n

X

i=1 ηi

z g_i00(z)

g_i0(z) +1

−1

! (5)

Since_|fi(z)| ≤Mi for alli=1, 2, . . . ,n, applying the General Schwarz Lemma, it results

fi(z) z

≤

M_i for alli=1, 2, . . . ,n. (6)

From (5) and (6), using that f_{i∈ B}(µ_i,α_i)and g_{i∈ N}(ρ_i)for alli=1, 2, . . . ,n, we obtain

zG00_n(z)

G0_n(z)

<

n

X

i=1

h

(γi−1)(2−αi)Mµi− 1

i +ηi(ρi−1)

i

=1−δ

This completes the proof of our theorem.

Letting n=1, γ₁ =γ, η₁ =η, M1 = M, µ1 =µ, α1 =α, ρ1= ρ, f1 = f and g1= gin

Theorem 2, we have

Corollary 3. Letγ∈R, γ >1, η∈R, η >0, the functions f ∈ B(µ,α), µ≥0,0≤α <1,

satisfying the condition|f(z)| ≤M ,(M ≥1)and g∈ N(ρ),ρ >1. If

(γ−1)(2₋α)Mµ−1+η(ρ−1)<1

then the integral operator

G₁(z) =

Z z

0

f(t)γ−1

(g0(t))ηdt

is in K(δ), whereδ=1+ (γ−1)(α−2)Mµ−1+η(1−ρ).

Corollary 4. Letγ_{i ∈}R, γi >1,ηi ∈R,ηi>0for all i=1, 2, . . . ,n, the functions fi ∈R(αi),

0 _≤ α_i < 1, satisfying the conditions |fi(z)| ≤ M , (M ≥ 1) and gi ∈ N(ρi), ρi > 1for all i=1, 2, . . . ,n. If

n

X

i=1

(γi−1)(2−αi)

1

M +ηi(ρi−1)

<1

then the integral operator Gndefined in(1)is in K(δ), where

δ=1₋

n

X

i=1

(γi−1)(2−αi)

1

M +ηi(ρi−1)

.

Lettingµ_i=1 and M_i=M for alli=1, 2, . . . ,nin Theorem 2, we have

Corollary 5. Letγi∈R,γi >1,ηi ∈R,ηi >0for all i=1, 2, . . . ,n, the functions fi∈S∗(αi),

0 ≤ αi < 1, satisfying the conditions |fi(z)| ≤ M , (M ≥ 1) and gi ∈ N(ρi), ρi > 1for all i=1, 2, . . . ,n. If

n

X

i=1

(γi−1)(2−αi) +ηi(ρi−1)

<1

then the integral operator G_ndefined in(1)is in K(δ), where

δ=1₋

n

X

i=1

(γi−1)(2−αi) +ηi(ρi−1)

.

Theorem 3. Letγi∈R,γi>1,ηi∈R,ηi>0for all i=1, 2, . . . ,n, the functions

f_i ∈ ρi −Sp("i), −1 ≤ "i ≤ 1, ρi > 0 and gi ∈ K D(µi,αi), 0 ≤ αi < 1, µi ≥ 0 for all i=1, 2, . . . ,n. If

0<

n

X

i=1

(1₋γ_i)"_i+η_i(1₋α_i)

≤1

then the integral operator G_ndefined in(1)is in K(δ), where

δ=1+

n

X

i=1

(γi−1)"i+ηi(αi−1)

.

Proof. Following the same steps as in Theorem 1, we obtain that

zG_n00(z)

G_n0(z) = n

X

i=1

(γi−1) z f_i0(z)

fi(z)

+

n

X

i=1 ηi

z g_i00(z)

g_i0(z)

and hence

zG00_n(z)

G0_n(z) +1= n

X

i=1

–

(γi−1)

‚

z f_i0(z)

fi(z)

−"i

Œ

+ (γi−1)"i

™

+

n

X

i=1

–

ηi

‚

z g00_i(z) g0_i(z) +1

Œ

−ηi

™

We calculate the real part from both terms of the above expression and obtain

Re

‚

zG_n00(z)

G_n0(z) +1

Œ

=

n

X

i=1

(γi−1)Re

‚

z f_i0(z)

fi(z)

−"i

Œ

+

n

X

i=1

(γi−1)"i

+

n

X

i=1 ηiRe

‚

z g00_i (z)

g0_i(z) +1

Œ

−

n

X

i=1 ηi+1.

(7)

From (7), using that f_i∈ρi−Sp("i)andgi ∈K D(µi,αi)for alli=1, 2, . . . ,n, we have

Re

‚

zG00_n(z)

G0_n(z) +1

Œ

≥

n

X

i=1

(γi−1)ρi

z f_i0(z)

fi(z)

−1 + n X

i=1

(γi−1)"i

+

n

X

i=1 ηi µi

z g_i00(z)

g_i0(z)

+αi

!

−

n

X

i=1 ηi+1.

Then

Re

‚

zG_n00(z)

G_n0(z) +1

Œ

≥

n

X

i=1

(γi−1)ρi

z f_i0(z)

fi(z)

−1 + n X

i=1

(γi−1)"i

+

n

X

i=1 ηiµi

z g_i00(z)

g_i0(z)

+ n X

i=1

ηi αi−1

+1.

(8)

Since(γ_i−1)ρ_i

z f_i0(z)

f_i(z) −1

>0 andη_iµ_i

z g00_i (z)

g0_i(z)

≥0 for alli=1, 2, . . . ,n, we obtain from (8) that

Re

‚

zG_n00(z) G_n0(z) +1

Œ

>

n

X

i=1

(γi−1)"i+ηi(αi−1)

+1=δ.

This completes the proof of our theorem.

Letting n= 1,γ₁ =γ, η₁ =η, ρ₁ =ρ, "₁ =", µ₁ = µ, α₁ = α, f1 = f and g1 = g in

Theorem 3, we have

Corollary 6. Letγ∈R,γ >1,η∈R,η >0, the functions f ∈ρ−Sp("),−1≤"≤1,ρ >0 and g_∈K D(µ,α),0_≤α <1,µ_≥0. If

0<(1₋γ)"+η(1₋α)_≤1

then the integral operator

G1(z) =

Z z

0

f(t)γ−1

(g0(t))ηdt

is in K(δ), whereδ=1+ (γ₋1)"+η(α₋1).

Corollary 7. Let the functions f _∈ρ₋S_p("),₋1_≤"_≤1,ρ >0and g_∈K D(µ,α),0_≤α <1,

µ≥0. If

0<1₋α₋"_≤1

then the integral operator

G(z) =

Z z

0

f(t)g0(t)dt is in K(δ), whereδ="+α.

References

[1] M Darus. Certain class of uniformly analytic functions. Acta Mathematica Academiae Pedagogicae Nyregyhaziensis, 24:354-358, 2008.

[2] B A Frasin and J M Jahangiri. A new and comprehensive class of analytic functions.

Annals of Oradea University-Mathematics Fascicola, XV:59-62, 2008.

[3] O Mayer.The functions theory of one variable complex. Bucuresti, 1981.

[4] A Mohammed, M Darus, and D Breaz, Some properties for certain integral operators.

Acta Universitatis Apulensis, 23:79-89, 2010.

[5] S Owa and H M Srivastava. Some generalized convolution properties associated with certain subclasses of analytic functions. Journal of Inequalities in Pure and Applied Math-ematics, 3(3):42:1-13, 2003.