We denote by P the class of the functions p which are analytic in U, p(0

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

a_kz^k,

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

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}, γ_i ∈C;fi∈ A;gi∈ P for all i∈ {1,2, ..., n}.

(3) G_β1,_β2, ...,_βn,γ1,γ2, ...,γn(z) = Z z

0 n

Y

i=1

f_i⁰(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

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

f(z) =e^iθ M R^mz^m, where θ is constant.

2. MAIN RESULTS

Theorem 1. Let the functions f_i be in the class B(µ_i, α_i), µ_i ≥ 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

|f_i(z)| ≤Mi (z∈U),

zg_i⁰(z) g_i(z)

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

δ= 1−

n

X

i=1

y_i >0, yi = 1

|β_i| (2−αi)Miµi−1

+ 1

+|γ_i−1|Ni

for all i∈ {1,2, ...., n}.

In these conditions the integral operator F_β1,_β2, ...,_βn,γ1,γ2, ...,γn(z) de- fined by (2) is in K(δ).

Proof. From (2) we obtain

F_β⁰₁,_β2, ...,_βn,γ1,γ2, ...,γn(z) =

n

Y

i=1

fi(z) z

¹

βi (gi(z))^γi−1 and

F_β⁰⁰₁,_β2, ...,_βn,γ1,γ2, ...,γn(z)

=

n

X

i=1



 1 β_i

fi(z) z

^1−βi

βi zf_i⁰(z)−fi(z)

z² (gi(z))^γi−1





n

Y

k=1 k6=i

fk(z) z

¹

βk (g_k(z))^γk−1

+

n

X

i=1

fi(z) z

¹

βi (γi−1) (gi(z))^γi−2g⁰_i(z)

! _n Y

k=1 k6=i

f_k(z) z

¹

βk (g_k(z))^γk−1.

After the calculus we obtain that (4)

zF_β⁰⁰

1,_β2, ...,_βn,_γ1,_γ2, ...,_γn(z) F_β⁰

1,_β2, ...,_βn,_γ1,_γ2, ...,_γn(z) =

n

X

i=1

1 β_i

zf_i⁰(z) f_i(z) −1

+ (γ_i−1)zg_i⁰(z) g_i(z)

. It follows from (4) that

(5)

zF_β⁰⁰

1,_β2, ...,_βn,_γ1,_γ2, ...,_γn(z) F_β⁰

1,_β2, ...,_βn,_γ1,_γ2, ...,_γn(z)

=

n

X

i=1

1 β_i

zf_i⁰(z) f_i(z) −1

+ (γ_i−1)zg_i⁰(z) g_i(z)

≤

n

X

i=1

1

|β_i|

zf_i⁰(z) fi(z)

+ 1

+|γ_i−1|

zg_i⁰(z) gi(z)

≤

n

X

i=1

1

|β_i|

f_i⁰(z) z

fi(z) µi

fi(z) z

µi−1

+ 1

!

+|γ_i−1|

zg⁰_i(z) gi(z)

! . Since

|f_i(z)| ≤Mi (z∈U),

zg_i⁰(z) gi(z)

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

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

(6)

zF_β⁰⁰

1,_β2, ...,_βn,γ1,γ2, ...,γn(z) F_β⁰

1,_β2, ...,_βn,γ1,γ2, ...,γn(z)

≤

n

X

i=1

1

|β_i|

f_i⁰(z) z

f_i(z) µi

M_i^µi−1+ 1

+|γ_i−1|N_i

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

zF_β⁰⁰

1,_β2, ...,_βn,_γ1,_γ2, ...,_γn(z) F_β⁰

1,_β2, ...,_βn,γ1,γ2, ...,γn(z)

≤

n

X

i=1

1

|β_i|

f_i⁰(z) z

fi(z) µi

−1

+ 1

M_i^µi−1+ 1

+|γ_i−1|N_i

≤

n

X

i=1

1

|β_i| (2−αi)Miµi−1

+ 1

+|γ_i−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 S^∗(αi), 0 ≤ αi < 1 and g_i ∈ P for all i ∈ {1,2, ..., n}. For any given 0 ≤ α_i < 1, M_i ≥ 1 and N_i ≥1 satisfying the conditions

|f_i(z)| ≤M_i (z∈U),

zg_i⁰(z) g_i(z)

≤N_i (z∈U)

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

n

X

i=1

yi >0,

y_i = 1

|β_i|(3−α_i) +|γ_i−1|N_i for all i∈ {1,2, ...., n}.

In these conditions, the integral operator F_β1,_β2, ...,_βn,γ1,γ2, ...,γn(z) de- fined by (2) is in K(δ).

Setting α1 =α2=...=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 f_i ∈ A, β_i complex numbers, βi 6= 0 for alli∈ {1,2, ..., n}andγ1=γ2 =...=γn= 1,we obtain the integral operator

F_β1,_β2, ...,_βn(z) = Z z

0 n

Y

i=1

f_i(t) t

¹

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

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

For any given λ_i ≥1, M_i≥1 satisfying the conditions (7)

zf_i⁰⁰(z) f_i⁰(z) −1

≤λ_i (z∈U),

zg⁰_i(z) gi(z)

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

δ= 1−

n

X

i=1

y_i >0, yi =|β_i|(λi+ 1) +|γ_i|Mi,

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

In these conditions, the integral operator G_β1,_β2, ...,_βn,_γ1,_γ2, ...,_γn(z)de- fined by (3) is in K(δ).

Proof. From (3) we obtain

G⁰_β1,β2, ...,βn,γ1,γ2, ...,γn(z) =

n

Y

i=1

f_i⁰(z)βi

(gi(z))^γi and

G⁰⁰_β1,_β2, ...,_βn,γ1,γ2, ...,γn(z)

=

n

X

i=1

βi f_i⁰(z)βi−1

f_i⁰⁰(z) (gi(z))^γi Yⁿ

k=1 k6=i

f_k⁰(z)β_k

(gk(z))^γk

+

n

X

i=1

f_i⁰(z)βi

γ_i(g_i(z))^γi−1g_i⁰(z)Yⁿ

k=1 k6=i

f_k⁰(z)βk

(g_k(z))^γk. After the calculus we obtain that

(8) zG⁰⁰_β

1,_β2, ...,_βn,γ1,γ2, ...,γn(z) G⁰_β

1,_β2, ...,_βn,γ1,γ2, ...,γn(z) =

n

X

i=1

βi

zf_i⁰⁰(z) f_i⁰(z) +γi

zg_i⁰(z) gi(z)

. It follows from (8) that

(9)

zG⁰⁰_β

1,_β2, ...,_βn,γ1,γ2, ...,γn(z) G⁰_β

1,β2, ...,βn,γ1,γ2, ...,γn(z)

=

n

X

i=1

β_izf_i⁰⁰(z)

f_i⁰(z) +γ_izg_i⁰(z) gi(z)

≤

n

X

i=1

|β_i|

zf_i⁰⁰(z) f_i⁰(z)

+|γ_i|

zg⁰_i(z) g_i(z)

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

zf_i⁰⁰(z) f_i⁰(z) −1

≤λi (z∈U),

zg_i⁰(z) gi(z)

≤Mi (z∈U).

Using the inequality (9) we obtain

zG⁰⁰_β1,β2, ...,βn,γ1,γ2, ...,γn(z) G⁰_β

1,_β2, ...,_βn,_γ1,_γ2, ...,_γn(z)

≤Pn i=1

|β_i|

zf_i⁰⁰(z) f_i⁰(z) −1

+ 1

+|γ_i|M_i

≤Pn

i=1(|β_i|(λ_i+ 1) +|γ_i|M_i)

= 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

G_α(z) = Z z

0

f⁰(t)α

dt studied in [7].

Remark 9.From (3) for β_i =α_i >0 for all i∈ {1,2, ..., n}and γ₁ =γ₂= ....=γ_n= 0,we obtain the integral operator

G_α1,_α2, ...,_αn(z) = Z z

0 n

Y

i=1

f_i⁰(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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[3] B.A. Frasin and M. Darus,On certain analytic univalent functions. Int. J. Math. Math.

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

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

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Received 3 October 2011 University of Pite¸sti, Department of Mathematics, Tˆargul 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