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FOR TWO INTEGRAL OPERATORS

LAURA STANCIU and DANIEL BREAZ Communicated by the former editorial board

In this paper, we consider two integral operators for analytic functionsfi(z) in the open unit diskU.The object of this paper is to prove the convexity for these integral operators.

AMS 2010 Subject Classification: Primary 30C45; Secondary 30C75.

Key words: analytic functions, univalent functions, general Schwarz lemma, in- tegral operators.

1. INTRODUCTION

Let U={z∈C:|z|<1}be the open unit disk and A denote the class of functions of the form

f(z) =z+

X

k=2

akzk,

which are analytic inUand satisfy the following usual normalization condition f(0) =f0(0)−1 = 0.

We denote by P the class of the functions p which are analytic in U, p(0) = 1 and Re{p(z)}>0,for allz∈U.

Let S be the subclass of Aconsisting of all univalent functions f inU. A function f belonging to S is said to be starlike of orderα if it satisfies

Re

zf0(z) f(z)

> α (z∈U) for someα(0≤α <1).

We denote by S(α) the subclass of A consisting of functions which are starlike of order α inU.

A function f belonging to S is said to be convex of order α if it satisfies Re

1 +zf00(z) f0(z)

> α (z∈U) for someα(0≤α <1).

MATH. REPORTS16(66),1(2014), 25–32

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We denote by K(α) the subclass of A consisting of functions which are convex of orderα inU.

A function f ∈ Ais said to be in the classR(α) if and only if Re f0(z)

> α, (z∈U). It is well known that K(α)⊂ S(α)⊂ S.

Frasin and Jahangiri [4] defined the family B(µ, α), µ≥0, 0≤α <1 so that it consists of functionsf ∈ B(µ, α) satisfying the condition

(1)

f0(z) z

f(z) µ

−1

<1−α, (z∈U).

The family B(µ, α) is a comprehensive class of analytic functions which includes various new classes of analytic univalent functions as well as some very well-known ones. Frasin and Darus in [3] introduce the special class B(2, α) = B(α). Other classes of univalent analytic functions are B(1, α) = S(α) and B(0, α) =R(α).

In this paper, we will obtain the order of convexity of the following integral operators

(2) Fβ1,β2, ...,βn,γ1,γ2, ...,γn(z) = Z z

0 n

Y

i=1

fi(t) t

1

βi (gi(t))γi−1dt

βi∈C\ {0}, γi ∈C;fi∈ A;gi∈ P for all i∈ {1,2, ..., n}.

(3) Gβ1,β2, ...,βn,γ1,γ2, ...,γn(z) = Z z

0 n

Y

i=1

fi0(t)βi

(gi(t))γidt βi, γi∈C;fi∈ A;gi ∈ P for all i∈ {1,2, ..., n}.

In order to prove our main results, we recall the following lemma

General Schwarz Lemma [6]. Let f the function regular in the disk UR ={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

Rm|z|m (z∈UR). The equality can hold only if

f(z) =e M Rmzm, where θ is constant.

(3)

2. MAIN RESULTS

Theorem 1. Let the functions fi be in the class B(µi, αi), µi ≥ 0, 0≤ αi < 1 and gi ∈ P for all i∈ {1,2, ..., n}. For any given µi ≥ 0, 0 ≤αi <1, Mi ≥1 and Ni≥1 satisfying the conditions

|fi(z)| ≤Mi (z∈U),

zgi0(z) gi(z)

≤Ni (z∈U) there exist numbers βi ∈C\ {0}, γi ∈C such that

δ= 1−

n

X

i=1

yi >0, yi = 1

i| (2−αi)Miµi−1

+ 1

+|γi−1|Ni

for all i∈ {1,2, ...., n}.

In these conditions the integral operator Fβ1,β2, ...,βn,γ1,γ2, ...,γn(z) de- fined by (2) is in K(δ).

Proof. From (2) we obtain

Fβ01,β2, ...,βn,γ1,γ2, ...,γn(z) =

n

Y

i=1

fi(z) z

1

βi (gi(z))γi−1 and

Fβ001,β2, ...,βn,γ1,γ2, ...,γn(z)

=

n

X

i=1

 1 βi

fi(z) z

1−βi

βi zfi0(z)−fi(z)

z2 (gi(z))γi−1

n

Y

k=1 k6=i

fk(z) z

1

βk (gk(z))γk−1

+

n

X

i=1

fi(z) z

1

βii−1) (gi(z))γi−2g0i(z)

! n Y

k=1 k6=i

fk(z) z

1

βk (gk(z))γk−1.

After the calculus we obtain that (4)

zFβ00

1,β2, ...,βn,γ1,γ2, ...,γn(z) Fβ0

1,β2, ...,βn,γ1,γ2, ...,γn(z) =

n

X

i=1

1 βi

zfi0(z) fi(z) −1

+ (γi−1)zgi0(z) gi(z)

. It follows from (4) that

(5)

zFβ00

1,β2, ...,βn,γ1,γ2, ...,γn(z) Fβ0

1,β2, ...,βn,γ1,γ2, ...,γn(z)

(4)

=

n

X

i=1

1 βi

zfi0(z) fi(z) −1

+ (γi−1)zgi0(z) gi(z)

n

X

i=1

1

i|

zfi0(z) fi(z)

+ 1

+|γi−1|

zgi0(z) gi(z)

n

X

i=1

1

i|

fi0(z) z

fi(z) µi

fi(z) z

µi−1

+ 1

!

+|γi−1|

zg0i(z) gi(z)

! . Since

|fi(z)| ≤Mi (z∈U),

zgi0(z) gi(z)

≤Ni (z∈U), applying the General Schwarz Lemma for the functions fi,we have

|fi(z)| ≤Mi|z|, (z∈U;i∈ {1,2, ..., n}). Therefore, from (5) we obtain

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zFβ00

1,β2, ...,βn,γ1,γ2, ...,γn(z) Fβ0

1,β2, ...,βn,γ1,γ2, ...,γn(z)

n

X

i=1

1

i|

fi0(z) z

fi(z) µi

Miµi−1+ 1

+|γi−1|Ni

. From (1) and (6), we have

zFβ00

1,β2, ...,βn,γ1,γ2, ...,γn(z) Fβ0

1,β2, ...,βn,γ1,γ2, ...,γn(z)

n

X

i=1

1

i|

fi0(z) z

fi(z) µi

−1

+ 1

Miµi−1+ 1

+|γi−1|Ni

n

X

i=1

1

i| (2−αi)Miµi−1

+ 1

+|γi−1|Ni

= 1−δ.

This completes the proof.

If we set µ1 = µ2 = ... = µn = 1 in Theorem 1, we can obtain the following interesting consequence of this theorem.

Corollary 2. Let the functions fi be in the class Si), 0 ≤ αi < 1 and gi ∈ P for all i ∈ {1,2, ..., n}. For any given 0 ≤ αi < 1, Mi ≥ 1 and Ni ≥1 satisfying the conditions

|fi(z)| ≤Mi (z∈U),

zgi0(z) gi(z)

≤Ni (z∈U)

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there exist numbers βi ∈C\ {0}, γi ∈C such that δ= 1−

n

X

i=1

yi >0,

yi = 1

i|(3−αi) +|γi−1|Ni for all i∈ {1,2, ...., n}.

In these conditions, the integral operator Fβ1,β2, ...,βn,γ1,γ2, ...,γn(z) de- fined by (2) is in K(δ).

Setting α12=...=αn= 0, δ= 0 and n= 1 in Corollary 2, we have Corollary 3. Let f ∈ A be starlike function in U and g ∈ P. For any given numbers M ≥1 and N ≥1 satisfying the conditions

|f(z)| ≤M (z∈U),

zg0(z) g(z)

≤N (z∈U) there exist numbers β ∈C\ {0}, γ∈Csuch that

1 = 3

|β|+N|γ−1|. In these conditions, the integral operator

Fβ,γ(z) = Z z

0

f(t) t

1

β

(g(t))γ−1dt is convex in U.

Remark 4. From (2) forn= 1, f1=f, β1

1 =α and γ1 = 1 we obtain the integral operator Kim-Merkes

Fα(z) = Z z

0

f(t) t

α

dt studied in [5].

Remark 5.In the relation (2) considering fi ∈ A, βi complex numbers, βi 6= 0 for alli∈ {1,2, ..., n}andγ12 =...=γn= 1,we obtain the integral operator

Fβ1,β2, ...,βn(z) = Z z

0 n

Y

i=1

fi(t) t

1

βi dt studied by D. Breaz and N. Breaz in [2].

(6)

Theorem 6. Let the functions fi ∈ Aand gi∈ P for all i∈ {1,2, ..., n}.

For any given λi ≥1, Mi≥1 satisfying the conditions (7)

zfi00(z) fi0(z) −1

≤λi (z∈U),

zg0i(z) gi(z)

≤Mi (z∈U) there exist numbers βi ∈C, γi ∈C such that

δ= 1−

n

X

i=1

yi >0, yi =|βi|(λi+ 1) +|γi|Mi,

i|, |γi|sufficiently small numbers for all i∈ {1,2, ...., n}.

In these conditions, the integral operator Gβ1,β2, ...,βn,γ1,γ2, ...,γn(z)de- fined by (3) is in K(δ).

Proof. From (3) we obtain

G0β1,β2, ...,βn,γ1,γ2, ...,γn(z) =

n

Y

i=1

fi0(z)βi

(gi(z))γi and

G00β1,β2, ...,βn,γ1,γ2, ...,γn(z)

=

n

X

i=1

βi fi0(z)βi−1

fi00(z) (gi(z))γi Yn

k=1 k6=i

fk0(z)βk

(gk(z))γk

+

n

X

i=1

fi0(z)βi

γi(gi(z))γi−1gi0(z)Yn

k=1 k6=i

fk0(z)βk

(gk(z))γk. After the calculus we obtain that

(8) zG00β

1,β2, ...,βn,γ1,γ2, ...,γn(z) G0β

1,β2, ...,βn,γ1,γ2, ...,γn(z) =

n

X

i=1

βi

zfi00(z) fi0(z) +γi

zgi0(z) gi(z)

. It follows from (8) that

(9)

zG00β

1,β2, ...,βn,γ1,γ2, ...,γn(z) G0β

1,β2, ...,βn,γ1,γ2, ...,γn(z)

=

n

X

i=1

βizfi00(z)

fi0(z) +γizgi0(z) gi(z)

n

X

i=1

i|

zfi00(z) fi0(z)

+|γi|

zg0i(z) gi(z)

From the hypothesis (7) of Theorem 6, we have

zfi00(z) fi0(z) −1

≤λi (z∈U),

zgi0(z) gi(z)

≤Mi (z∈U).

(7)

Using the inequality (9) we obtain

zG00β1,β2, ...,βn,γ1,γ2, ...,γn(z) G0β

1,β2, ...,βn,γ1,γ2, ...,γn(z)

≤Pn i=1

i|

zfi00(z) fi0(z) −1

+ 1

+|γi|Mi

≤Pn

i=1(|βi|(λi+ 1) +|γi|Mi)

= 1−δ.

This completes the proof of our theorem.

Setting n= 1 in Theorem 6 we obtain

Corollary 7. Let the functions f ∈ A and g∈ P. For any givenλ≥1 and M ≥1 satisfying the conditions

zf00(z) f0(z) −1

≤λ (z∈U),

zg0(z) g(z)

≤M (z∈U) there exist numbers β ∈C, γ∈Csuch that

δ= 1−(|β|(λ+ 1) +|γ|M)>0 and

(|β|(λ+ 1) +|γ|M)<1 for |β|, |γ|sufficiently small numbers.

In these conditions, the integral operator G(z) =

Z z 0

f0(t)β

(g(t))γdt is in K(δ).

Remark 8. From (3) forn= 1, β1 =α, f1 =f ∈ Aand γ1 = 0,we obtain the integral operator

Gα(z) = Z z

0

f0(t)α

dt studied in [7].

Remark 9.From (3) for βii >0 for all i∈ {1,2, ..., n}and γ12= ....=γn= 0,we obtain the integral operator

Gα1,α2, ...,αn(z) = Z z

0 n

Y

i=1

fi0(t)αi

dt studied in [1].

Acknowledgments. This work was partially supported by the strategic project POS- DRU 107/1.5/S/77265, inside POSDRU Romania 2007-2013 co-financed by the Euro- pean Social Fund-Investing in People.

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REFERENCES

[1] D. Breaz, Certain integral operators on the classesM(βi)andNi). J. Inequal. Appl.

(2008), 1–4.

[2] D. Breaz and N. Breaz, Two Integral Operators. Stud. Univ. Babe¸s-Bolyai Ser. Math.

3(2002), 13–21.

[3] B.A. Frasin and M. Darus,On certain analytic univalent functions. Int. J. Math. Math.

Sci.25(5) (2001), 305–310.

[4] B.A. Frasin and J. Jahangiri,A new and comprehensive class of analytic functions. Anal.

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

[6] Z. Nehari,Conformal mapping. McGraw-Hill Book Comp., New York, 1952.

[7] N.N. Pascu and V. Pescar, On the integral operators of Kim-Merkes and Pfaltzgraff.

Mathematica32(55) (1990),2, 185–192.

Received 3 October 2011 University of Pite¸sti, Department of Mathematics, argul din Vale Str., No.1, 110040,

Pite¸sti, Arge¸s, Romˆania laura stanciu 30@yahoo.com

“1 Decembrie 1918” University of Alba Iulia,

Department of Mathematics, Str. N. Iorga, 510000, No. 11-13,

Alba Iulia, Romˆania dbreaz@uab.ro

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