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_{k}z^{k},

which are analytic inUand satisfy the following usual normalization condition
f(0) =f^{0}(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

zf^{0}(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 +zf^{00}(z)
f^{0}(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 f^{0}(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)

f^{0}(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))^{γ}^{i}^{−1}dt

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

(3) G_{β}_{1},_{β}_{2}, ...,_{β}_{n},γ1,γ2, ...,γn(z) =
Z z

0 n

Y

i=1

f_{i}^{0}(t)βi

(gi(t))^{γ}^{i}dt
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^{m}z^{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_{i}^{0}(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_{β}_{1},_{β}_{2}, ...,_{β}_{n},γ1,γ2, ...,γn(z) de-
fined by (2) is in K(δ).

Proof. From (2) we obtain

F_{β}^{0}_{1},_{β}_{2}, ...,_{β}_{n},γ1,γ2, ...,γn(z) =

n

Y

i=1

fi(z) z

^{1}

βi (gi(z))^{γ}^{i}^{−1}
and

F_{β}^{00}_{1},_{β}_{2}, ...,_{β}_{n},γ1,γ2, ...,γn(z)

=

n

X

i=1

1
β_{i}

fi(z) z

^{1−}^{βi}

βi zf_{i}^{0}(z)−fi(z)

z^{2} (gi(z))^{γ}^{i}^{−1}

n

Y

k=1 k6=i

fk(z) z

^{1}

βk (g_{k}(z))^{γ}^{k}^{−1}

+

n

X

i=1

fi(z) z

^{1}

βi (γi−1) (gi(z))^{γ}^{i}^{−2}g^{0}_{i}(z)

! _{n}
Y

k=1 k6=i

f_{k}(z)
z

^{1}

βk (g_{k}(z))^{γ}^{k}^{−1}.

After the calculus we obtain that (4)

zF_{β}^{00}

1,_{β}_{2}, ...,_{β}_{n},_{γ}_{1},_{γ}_{2}, ...,_{γ}_{n}(z)
F_{β}^{0}

1,_{β}_{2}, ...,_{β}_{n},_{γ}_{1},_{γ}_{2}, ...,_{γ}_{n}(z) =

n

X

i=1

1
β_{i}

zf_{i}^{0}(z)
f_{i}(z) −1

+ (γ_{i}−1)zg_{i}^{0}(z)
g_{i}(z)

. It follows from (4) that

(5)

zF_{β}^{00}

1,_{β}_{2}, ...,_{β}_{n},_{γ}_{1},_{γ}_{2}, ...,_{γ}_{n}(z)
F_{β}^{0}

1,_{β}_{2}, ...,_{β}_{n},_{γ}_{1},_{γ}_{2}, ...,_{γ}_{n}(z)

=

n

X

i=1

1
β_{i}

zf_{i}^{0}(z)
f_{i}(z) −1

+ (γ_{i}−1)zg_{i}^{0}(z)
g_{i}(z)

≤

n

X

i=1

1

|β_{i}|

zf_{i}^{0}(z)
fi(z)

+ 1

+|γ_{i}−1|

zg_{i}^{0}(z)
gi(z)

≤

n

X

i=1

1

|β_{i}|

f_{i}^{0}(z)
z

fi(z) µi

fi(z) z

µi−1

+ 1

!

+|γ_{i}−1|

zg^{0}_{i}(z)
gi(z)

! . Since

|f_{i}(z)| ≤Mi (z∈U),

zg_{i}^{0}(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_{β}^{00}

1,_{β}_{2}, ...,_{β}_{n},γ1,γ2, ...,γn(z)
F_{β}^{0}

1,_{β}_{2}, ...,_{β}_{n},γ1,γ2, ...,γn(z)

≤

n

X

i=1

1

|β_{i}|

f_{i}^{0}(z)
z

f_{i}(z)
µi

M_{i}^{µ}^{i}^{−1}+ 1

+|γ_{i}−1|N_{i}

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

zF_{β}^{00}

1,_{β}_{2}, ...,_{β}_{n},_{γ}_{1},_{γ}_{2}, ...,_{γ}_{n}(z)
F_{β}^{0}

1,_{β}_{2}, ...,_{β}_{n},γ1,γ2, ...,γn(z)

≤

n

X

i=1

1

|β_{i}|

f_{i}^{0}(z)
z

fi(z) µi

−1

+ 1

M_{i}^{µ}^{i}^{−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_{i}^{0}(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_{β}_{1},_{β}_{2}, ...,_{β}_{n},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),

zg^{0}(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−1}dt
is convex in U.

Remark 4. From (2) forn= 1, f_{1}=f, _{β}^{1}

1 =α and γ_{1} = 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_{β}_{1},_{β}_{2}, ...,_{β}_{n}(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_{i}^{00}(z)
f_{i}^{0}(z) −1

≤λ_{i} (z∈U),

zg^{0}_{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_{β}_{1},_{β}_{2}, ...,_{β}_{n},_{γ}_{1},_{γ}_{2}, ...,_{γ}_{n}(z)de-
fined by (3) is in K(δ).

Proof. From (3) we obtain

G^{0}_{β}_{1},β2, ...,βn,γ1,γ2, ...,γn(z) =

n

Y

i=1

f_{i}^{0}(z)βi

(gi(z))^{γ}^{i}
and

G^{00}_{β}_{1},_{β}_{2}, ...,_{β}_{n},γ1,γ2, ...,γn(z)

=

n

X

i=1

βi f_{i}^{0}(z)βi−1

f_{i}^{00}(z) (gi(z))^{γ}^{i}
Y^{n}

k=1 k6=i

f_{k}^{0}(z)β_{k}

(gk(z))^{γ}^{k}

+

n

X

i=1

f_{i}^{0}(z)βi

γ_{i}(g_{i}(z))^{γ}^{i}^{−1}g_{i}^{0}(z)Y^{n}

k=1 k6=i

f_{k}^{0}(z)βk

(g_{k}(z))^{γ}^{k}.
After the calculus we obtain that

(8) zG^{00}_{β}

1,_{β}_{2}, ...,_{β}_{n},γ1,γ2, ...,γn(z)
G^{0}_{β}

1,_{β}_{2}, ...,_{β}_{n},γ1,γ2, ...,γn(z) =

n

X

i=1

βi

zf_{i}^{00}(z)
f_{i}^{0}(z) +γi

zg_{i}^{0}(z)
gi(z)

. It follows from (8) that

(9)

zG^{00}_{β}

1,_{β}_{2}, ...,_{β}_{n},γ1,γ2, ...,γn(z)
G^{0}_{β}

1,β2, ...,βn,γ1,γ2, ...,γn(z)

=

n

X

i=1

β_{i}zf_{i}^{00}(z)

f_{i}^{0}(z) +γ_{i}zg_{i}^{0}(z)
gi(z)

≤

n

X

i=1

|β_{i}|

zf_{i}^{00}(z)
f_{i}^{0}(z)

+|γ_{i}|

zg^{0}_{i}(z)
g_{i}(z)

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

zf_{i}^{00}(z)
f_{i}^{0}(z) −1

≤λi (z∈U),

zg_{i}^{0}(z)
gi(z)

≤Mi (z∈U).

Using the inequality (9) we obtain

zG^{00}_{β}_{1},β2, ...,βn,γ1,γ2, ...,γn(z)
G^{0}_{β}

1,_{β}_{2}, ...,_{β}_{n},_{γ}_{1},_{γ}_{2}, ...,_{γ}_{n}(z)

≤Pn i=1

|β_{i}|

zf_{i}^{00}(z)
f_{i}^{0}(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

zf^{00}(z)
f^{0}(z) −1

≤λ (z∈U),

zg^{0}(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

f^{0}(t)β

(g(t))^{γ}dt
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

f^{0}(t)α

dt studied in [7].

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

G_{α}_{1},_{α}_{2}, ...,_{α}_{n}(z) =
Z z

0 n

Y

i=1

f_{i}^{0}(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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[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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Mathematica32(55) (1990),2, 185–192.

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