EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 5, 2020, 1285-1299
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Special Issue Dedicated to
Professor Hari M. Srivastava

On the Occasion of his 80th Birthday

Some univalence conditions of a certain general integral
operator

Camelia Bărbatu1,∗, Daniel Breaz2

1 Department of Mathematics, Babe Bolyai University, Cluj-Napoca, Romania
2 Department of Exact Science and Engineering, 1 Decembrie 1918 University, Alba Iulia,
Romania

Abstract. For some classes of analytic functions f , g, h and k in the open unit disk U, we
consider the general integral operator Tn, that was introduced in a recent work [1] and we obtain
new conditions of univalence for this integral operator. The key tools in the proofs of our results are
the Pascu’s and the Pescar’s univalence criteria, as well as the Mocanu’s and erb’s Lemma. Some
corollaries of the main results are also considered. Relevant connections of the results presented
here with various other known results are briefly indicated.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Integral operators, analytic and univalent functions, open unit disk,
univalence conditions, Schwarz Lemma

1. Introduction and preliminaries

Let A denote the class of the functions of the form:

f(z) = z +

∞∑
n=2

anz
n, (1)

which are analytic in the open unit disk

U = {z ∈ C :| z |< 1}

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3679

Email addresses: camipode@yahoo.com (C. Bărbatu), dbreaz@uab.ro (D. Breaz)

https://www.ejpam.com 1285 c© 2020 EJPAM All rights reserved.



C. Bărbatu, D. Breaz / Eur. J. Pure Appl. Math, 13 (5) (2020), 1285-1299 1286

and satisfy the following usual normalization conditions:

f(0) = f
′
(0)− 1 = 0,

C being the set of complex numbers.
We denote by S the subclass of A consisting of functions f ∈ A, which are univalent

in U.
A function f ∈ A said to be in the class S∗ (α) of starlike functions of order α

(0 ≤ α < 1) in U, if it satisfies the following inequality:

Re

[
zf
′
(z)

f(z)

]
> α, z ∈ U.

.
A function f ∈ A is said to belong to the class R(λ), 0 ≤ λ < 1, if

Re
[
f
′
(z)
]
> λ, z ∈ U.

Frasin and Jahangiri [14] studied the class B (µ, λ), µ ≥ 0, 0 ≤ λ < 1, which consists
of functions f ∈ A that satisfy the following conditions:∣∣∣∣f ′(z)( z

f(z)

)µ
− 1

∣∣∣∣ < 1− λ, z ∈ U. (2)

.
This class B (µ, λ) is a comprehensive class of normalized analytic functions in U. For

instance, we have B (1, λ) = S∗(λ), B (0, λ) = R(λ) and B (2, λ) = B(λ). In particular, the
analytic and univalent function class B(λ) was studied by Frasin and Darus [13].

We consider the integral operator

Tn(z) =

{
δ

∫ z

0
tδ−1

n∏
i=1

[(
fi(t)

t

)αi−1 (
g′i(t)

)βi (hi(t)
ki(t)

)γi (hi′(t))
ki
′(t)

)δi]
dt

} 1
δ

, (3)

where fi, gi, hi, ki are analytic in U and αi, βi, γi, δi ∈ C for all i = 1, n, n ∈ N\{0}, δ ∈ C,
with Reδ > 0.

Remark 1. The integral operator Tn defined by (3), introduced by Bărbatu and Breaz in
the paper [1] is a general integral operator of Pfaltzgraff, Kim-Merkes and Oversea types
which extends also the other operators as follows:

i) For n = 1, δ = 1, α1 − 1 = α1 and β1 = γ1 = δ1 = 0 we obtain the integral operator
which was studied by Kim-Merkes [15].

Fα(z) =

∫ z

0

(
f(t)

t

)α
dt,



C. Bărbatu, D. Breaz / Eur. J. Pure Appl. Math, 13 (5) (2020), 1285-1299 1287

ii) For n = 1, δ = 1 and α1 − 1 = γ1 = δ1 = 0 we obtain the integral operator which was
studied by Pfaltzgraff [27].

Gα(z) =

∫ z

0

(
f ′(t)

)α
dt,

iii) For αi − 1 = αi and βi = γi = δi = 0 we obtain the integral operator which was
defined and studied by D. Breaz and N. Breaz [3].

Dn(z) =

[
δ

∫ z

0
tδ−1

n∏
i=1

(
fi(t)

t

)αi
dt

] 1
δ

,

this integral operator is a generalization of the integral operator introduced by Pascu and
Pescar [23].

iv) For αi − 1 = γi = δi = 0 we obtain the integral operator which was defined and
studied by D. Breaz, Owa and N. Breaz [4]

In(z) =

[
δ

∫ z

0
tδ−1

n∏
i=1

[
f ′i(t)

]αi dt

] 1
δ

,

this integral operator is a generalization of the integral operator introduced by Pescar and
Owa in [26] .

v) For αi − 1 = αi and γi = δi = 0 we obtain the integral operator which was studied
by Ularu in [28]

In(z) =

[
δ

∫ z

0
tδ−1

n∏
i=1

(
fi(t)

t

)αi (
gi
′(t)
)βi dt

] 1
δ

.

vi) For αi − 1 = βi = 0, ki(z) = z and k′i(z) = 1 we obtain the integral operator which
was defined and studied by Pescar [25]

Fn(z) =

[
δ

∫ z

0
tδ−1

n∏
i=1

(
fi(t)

t

)αi (
fi
′(t)
)βi dt

] 1
δ

,

this integral operator is a generalization of the integral operator introduced by Frasin in
[12] and by Oversea in [20].

vii) For αi − 1 = βi = 0 we obtain the integral operator which was defined and studied
by Pescar [25]

In(z) =

δ ∫ z

0
tδ−1

n∏
i=1

(
fi(t)

gi(t)

)γi (f ′i (t)
g
′
i(t)

)δi
dt

 1
δ

.

viii) For δ = 1, αi− 1 = γi = 0, βi = δi and hi(z) = z2

2 we obtain the integral operator
which was defined and studied by Bucur and Breaz in [6]

In(z) =

∫ z

0

n∏
i=1

[
tg
′
i(t)

k
′
i(t)

]βi
dt,



C. Bărbatu, D. Breaz / Eur. J. Pure Appl. Math, 13 (5) (2020), 1285-1299 1288

this integral operator is a generalization of the integral operator introduced by Bucur, An-
drei and Breaz in [10] and [11].

xi) For δ = 1, αi−1 = δi = 0, βi = γi and hi(z) = fi(z) we obtain the integral operator
which was defined and studied by Nguyen, Oprea and Breaz in [18]

Hn,α(z) =

∫ z

0

n∏
i=1

(
fi(t)

hi(t)
g
′
i(t)

)αi
dt.

Thus, the integral operator Tn, introduced here by the formula (3), can be considered
as an extension and a generalization of these operators above mentioned.

The following univalence condition was derived by Pascu.

Theorem 1. (Pascu [22]) Let δ ∈ C with Reδ > 0. If f ∈ A satisfies

1− |z|2Reδ

Reδ

∣∣∣∣zf ′′(z)f ′(z)

∣∣∣∣ ≤ 1,

for all z ∈ U, then, for any complex γ with Reγ ≥ Reδ, the integral operator

Fγ(z) =

(
γ

∫ z

0
tγ−1f ′(t)dt

) 1
γ

,

is in the class S.

Pescar, on the other hand, proved another univalent condition asserted by Theorem 2.

Theorem 2. (Pescar [25]) Let γ be complex number, Reγ > 0 and c a complex number,
|c| ≤ 1, c 6= −1, and f ∈ A, f(z) = z + a2z

2 + .... If∣∣∣∣c |z|2γ +
(

1− |z|2γ
) zf ′′(z)
γf ′(z)

∣∣∣∣ ≤ 1,

for all z ∈ U, then the integral operator

Fγ(z) =

(
γ

∫ z

0
tγ−1f ′(t)dt

)frac1γ
,

is in the class S.

Mocanu and erb proved the next Theorem.

Theorem 3. (Mocanu - erb [17]) Let M0 = 1, 5936... the positive solution of equation

(2−M) eM = 2. (4)

If f ∈ A and ∣∣∣∣f ′′(z)f ′(z)

∣∣∣∣ ≤M0,

for z ∈ U, then ∣∣∣∣zf ′(z)f(z)
− 1

∣∣∣∣ ≤ 1, (z ∈ U)

The bound M0 is sharp.



C. Bărbatu, D. Breaz / Eur. J. Pure Appl. Math, 13 (5) (2020), 1285-1299 1289

Finally, in our present investigation, we shall also need the familiar Schwarz Lemma.

Lemma 1. ( General Schwarz Lemma [16]) Let f be the function regular in the disk
UR = {z ∈ C : |z| < R,R > 0} with |f(z)| < M for a fixed number M > 0 fixed. If f(z)
has one zero with multiplicity order bigger than a positive integer m for z = 0, then

|f(z)| ≤ M

Rm
zm, z ∈ UR.

The equality for z 6= 0 can hold only if

f(z) = eiθ
M

Rm
zm,

where θ is constant.

The problem of univalence for some generalized integral operators using functions from
the class B (µ, λ) were recently obtained in papers[5], [7],[8], [11], [19].

2. Main results

Our main results give sufficient conditions for the general integral operator Tn defined
by (3) to be univalent in the open disk U.

Theorem 4. Let δ, γ, αi, βi, γi, δi ∈ C, c = Reγ > 0 and Mi, Ni, Pi, Qi, Ri, Si ≥ 1,
i = 1, n, such that

(2c+ 1)
2c+1
2c

n∑
i=1

{
|αi − 1|

[
1 + (2− λi)Mµi−1

i

]
+ |γi|

[
2 + (2− ηi)P νi−1i

]}
+

+ (2c+ 1)
2c+1
2c

n∑
i=1

|γi| (2− ρi)Qθi−1i +2c

n∑
i=1

[|βi|Ni + |δi| (Ri + Si)] ≤ c (2c+ 1)
2c+1
2c . (5)

If fi ∈ B (µi, λi) , gi ∈ A, hi ∈ B (νi, ηi), ki ∈ B (θi, ρi), satisfies

|fi (z)| < Mi,

∣∣∣∣∣g
′′
i (z)

g
′
i(z)

∣∣∣∣∣ ≤ Ni, |hi (z)| < Pi, |ki (z)| < Qi,

∣∣∣∣∣h
′′
i (z)

h
′
i(z)

∣∣∣∣∣ ≤ Ri,
∣∣∣∣∣k
′′
i (z)

k
′
i(z)

∣∣∣∣∣ ≤ Si,
for all z ∈ U, i = 1, n, then for every δ, Reδ ≥ Reγ, the function Tn, defined by (3) is in
the class S.

Proof. Let us define the function

Tn (z) =

∫ z

0

n∏
i=1

[(
fi(t)

t

)αi−1
·
(
g′i(t)

)βi · (hi (t)

ki(t)

)γi
·
(
hi
′ (t)

ki
′(t)

)δi]
dt,

for all fi, gi, hi, ki ∈ A, i = 1, n.



C. Bărbatu, D. Breaz / Eur. J. Pure Appl. Math, 13 (5) (2020), 1285-1299 1290

The function Tn is regular in U and satisfies the following normalization condition
Tn(0 ) = T

′
n(0 )− 1 = 0 .

After we calculate the first-order and second-order derivatives, we obtain

zT ′′n (z)

T ′n(z)
=

n∑
i=1

[
(αi − 1)

(
zf ′i(z)

fi(z)
− 1

)
+ βi

zg′′i (z)

g′i(z)

]
+

+

n∑
i=1

[
γi

(
zh′i(z)

hi(z)
− zk′i(z)

ki(z)

)
+ δi

(
zh′′i (z)

h′i(z)
− zk′′i (z)

k′i(z)

)]
.

Therefore ∣∣∣∣zT ′′n (z)

T ′n(z)

∣∣∣∣ ≤ n∑
i=1

(
|αi − 1|

∣∣∣∣zf ′i(z)fi(z)
− 1

∣∣∣∣+ |βi|
∣∣∣∣zgi′′(z)gi′(z)

∣∣∣∣)+

+

n∑
i=1

{
|γi|
[(∣∣∣∣zh′i(z)hi(z)

− 1

∣∣∣∣)+

(∣∣∣∣zk′i(z)ki(z)
− 1

∣∣∣∣)]+ |δi|
(∣∣∣∣zh′′i (z)h′i(z)

∣∣∣∣+

∣∣∣∣zk′′i (z)

k′i(z)

∣∣∣∣)} . (6)

Thus, clearly, we find from this last inequality (6) that

1− |z|2c

c

∣∣∣∣zT ′′n (z)

T ′n(z)

∣∣∣∣ ≤ 1− |z|2c

c

n∑
i=1

[
|αi − 1|

(∣∣∣∣zf ′i(z)fi(z)

∣∣∣∣+ 1

)
+ |βi|

∣∣∣∣∣zg
′′
i (z)

gi(z)

∣∣∣∣∣
]

+

+
1− |z|2c

c

n∑
i=1

{
|γi|
(∣∣∣∣zh′i(z)hi(z)

∣∣∣∣+

∣∣∣∣zk′i(z)ki(z)

∣∣∣∣+ 2

)
+ |δi|

(∣∣∣∣zh′′i (z)h′i(z)

∣∣∣∣+

∣∣∣∣zk′′i (z)

k′i(z)

∣∣∣∣)} ≤
≤ 1− |z|2c

c

n∑
i=1

[
|αi − 1|

(∣∣∣∣f ′i (z)( z

fi(z)

)µi∣∣∣∣ ∣∣∣∣fi(z)z

∣∣∣∣µi−1 + 1

)
+ |βi| |z|

∣∣∣∣zg′′i (z)

g′i(z)

∣∣∣∣
]

+

+
1− |z|2c

c

n∑
i=1

|γi|

(∣∣∣∣h′i(z)( z

hi(z)

)νi∣∣∣∣ ∣∣∣∣hi(z)z

∣∣∣∣νi−1 +

∣∣∣∣∣k′i(z)
(

z

ki(z)

)θi∣∣∣∣∣
∣∣∣∣ki(z)z

∣∣∣∣θi−1 + 2

)
+

+
1− |z|2c

c

n∑
i=1

|δi|
(
|z|
∣∣∣∣h′′i (z)h′i(z)

∣∣∣∣+ |z|
∣∣∣∣k′′i (z)

k′i(z)

∣∣∣∣) .
By applying the General Schwarz Lemma to the functions fi, hi, ki, i = 1, n we obtain

|fi (z)| ≤Mi |z| , |hi (z)| ≤ Pi |z| , |ki (z)| ≤ Qi |z| .

Next, using the hypothesis, we obtain:

1− |z|2c

c

∣∣∣∣zT ′′n (z)

T ′n(z)

∣∣∣∣ ≤ 1− |z|2c

c

n∑
i=1

|αi − 1|
(∣∣∣∣f ′i (z)( z

fi(z)

)µi
− 1

∣∣∣∣+ 1

)
Mµi−1
i +



C. Bărbatu, D. Breaz / Eur. J. Pure Appl. Math, 13 (5) (2020), 1285-1299 1291

+
1− |z|2c

c

n∑
i=1

{
|βi| |z|Ni + |γi|

[
(2− ηi)P νi−1i + (2− ρi)Qθi−1i + 2

]}
+

+
1− |z|2c

c

n∑
i=1

{|δi| |z| (Ri + Si)} . (7)

Since

max
|z|≤1

(
1− |z|2c

)
|z|

c
=

2

(2c+ 1)
2c+1
2c

,

we obtain

1− |z|2c

c

∣∣∣∣zT ′′n (z)

T ′n(z)

∣∣∣∣ ≤ 1

c

n∑
i=1

{
|αi − 1|

[
1 + (2− λi)Mµi−1

i

]
+ |γi|

[
1 + (2− ηi)P νi−1i

]}
+

+
1

c

n∑
i=1

|γi|
[
1 + (2− ρi)Qθi−1i

]
+

2

(2c+ 1)
2c+1
2c

n∑
i=1

[|βi|Ni + |δi| (Ri + Si)] . (8)

If we make use of (5), the last inequality yields

1− |z|2c

c

∣∣∣∣zT ′′n (z)

T ′n(z)

∣∣∣∣ ≤ 1

for all z ∈ U, i = 1, n.
Finally, we apply Theorem 1, we conclude that, the general integral operator Tn given

by (3) is in the class S.

Theorem 5. Let c, δ, αi, βi, γi, δi ∈ C, Reδ > 0 and Mi, Ni, Pi, Qi, Ri, Si ≥ 1, i = 1, n.
Suppose that fi ∈ B (µi, λi), gi ∈ A, hi ∈ B (νi, ηi), ki ∈ B (θi, ρi), satisfies

|fi (z)| < Mi,

∣∣∣∣∣zg
′′
i (z)

g
′
i(z)

∣∣∣∣∣ < Ni, |hi (z)| < Pi, |ki (z)| < Qi,

∣∣∣∣∣zh
′′
i (z)

h
′
i(z)

∣∣∣∣∣ < Ri,

∣∣∣∣∣zk
′′
i (z)

k
′
i(z)

∣∣∣∣∣ < Si,

for all z ∈ U, i = 1, n.
If

Reδ ≥
n∑
i=1

{
|αi − 1|

[
(2− λi)Mµi−1

i + 1
]

+ |βi|Ni

}
+

+

n∑
i=1

{
|γi|
[
(2− ηi)P νi−1i + (2− ρi)Qθi−1i + 2

]
+ |δi| (Ri + Si)

}
(9)

and

|c| ≤ 1− 1

Reδ

n∑
i=1

{
|αi − 1|

[
(2− λi)Mµi−1

i + 1
]

+ |βi|Ni

}
−



C. Bărbatu, D. Breaz / Eur. J. Pure Appl. Math, 13 (5) (2020), 1285-1299 1292

− 1

Reδ

n∑
i=1

{
|γi|
[
(2− ηi)P νi−1i + (2− ρi)Qθi−1i + 2

]
+ |δi| (Ri + Si)

}
(10)

for all z ∈ U, i = 1, n, then the function Tn, defined by (3) is in the class S.

Proof. Just as in the proof of Theorem 2.1, we have∣∣∣∣zT ′′n (z)

T ′n(z)

∣∣∣∣ ≤ n∑
i=1

(
|αi − 1|

∣∣∣∣zf ′i(z)fi(z)
− 1

∣∣∣∣+ |βi|
∣∣∣∣zgi′′(z)gi′(z)

∣∣∣∣)+

+

n∑
i=1

{
|γi|
[(∣∣∣∣zh′i(z)hi(z)

− 1

∣∣∣∣)+

(∣∣∣∣zk′i(z)ki(z)
− 1

∣∣∣∣)]+ |δi|
(∣∣∣∣zh′′i (z)h′i(z)

∣∣∣∣+

∣∣∣∣zk′′i (z)

k′i(z)

∣∣∣∣)} .
So, for a given constant c ∈ C, we obtain∣∣∣∣c |z|2Reδ +

(
1−

∣∣∣z2δ∣∣∣) zT ′′n (z)

δT ′n(z)

∣∣∣∣ ≤ |c|+ 1

|δ|

n∑
i=1

[
|αi − 1|

(∣∣∣∣zf ′i(z)fi(z)

∣∣∣∣+ 1

)
+ |βi|

∣∣∣∣∣zg
′′
i (z)

gi(z)

∣∣∣∣∣
]

+

+
1

|δ|

n∑
i=1

{
|γi|
[(∣∣∣∣zh′i(z)hi(z)

∣∣∣∣+ 1

)
+

(∣∣∣∣zk′i(z)ki(z)

∣∣∣∣+ 1

)]
+ |δi|

(∣∣∣∣zh′′i (z)h′i(z)

∣∣∣∣+

∣∣∣∣zk′′i (z)

k′i(z)

∣∣∣∣)} ≤
≤ |c|+ 1

|δ|

n∑
i=1

|αi − 1|

(∣∣∣∣f ′i (z)( z

fi(z)

)µi∣∣∣∣ ∣∣∣∣fi(z)z

∣∣∣∣µi−1 + 1

)
+

+
1

|δ|

n∑
i=1

[
|β|
∣∣∣∣zg′′i (z)

g′i(z)

∣∣∣∣+ |γi|

(∣∣∣∣h′i(z)( z

hi(z)

)µi∣∣∣∣ ∣∣∣∣hi(z)z

∣∣∣∣µi−1 + 1

)]
+

+
1

|δ|

n∑
i=1

{
|γi|

(∣∣∣∣k′i(z)( z

ki(z)

)νi∣∣∣∣ ∣∣∣∣ki(z)z

∣∣∣∣νi−1 + 1

)
+ |δi|

(∣∣∣∣zh′′i (z)h′i(z)

∣∣∣∣+

∣∣∣∣zk′′i (z)

k′i(z)

∣∣∣∣)
}
. (11)

Now, applying the General Schwarz Lemma to the functions fi, hi, ki, i = 1, n we
obtain

|fi (z)| ≤Mi |z| , |hi (z)| ≤ Pi |z| , |ki (z)| ≤ Qi |z| , (12)

Using the hypothesis and (12) in inequality (11), we have∣∣∣∣c |z|2Reδ +
(

1−
∣∣∣z2δ∣∣∣) zT ′′n (z)

δT ′n(z)

∣∣∣∣ ≤
≤ |c|+ 1

|δ|

n∑
i=1

|αi − 1|
[(∣∣∣∣f ′i (z)( z

fi(z)

)µi
− 1

∣∣∣∣+ 1

)
Mµi−1
i + 1

]
+

+
1

|δ|

n∑
i=1

{
|γi|
[(∣∣∣∣h′i(z)( z

hi(z)

)νi
− 1

∣∣∣∣+ 1

)
P νi−1i + 1

]
+ |βi|Ni

}



C. Bărbatu, D. Breaz / Eur. J. Pure Appl. Math, 13 (5) (2020), 1285-1299 1293

+
1

|δ|

n∑
i=1

{
|γi|

[(∣∣∣∣∣k′i(z)
(

z

ki(z)

)θi
− 1

∣∣∣∣∣+ 1

)
Qθi−1i + 1

]
+ |δi| (Ri + Si)

}
≤

≤ |c|+ 1

Reδ

n∑
i=1

{
|αi − 1|

[
(2− λi)Mµi−1

i + 1
]

+ |βi|Ni

}
+

+
1

Reδ

n∑
i=1

{
|γi|
[
(2− ηi)P νi−1i + 1 + (2− ρi)Qθi−1i + 1

]
+ |δi| (Ri + Si)

}
.

Finally, by applying Theorem 2 to the function Tn, we deduce that function Tn given by
(3) is in the class S.

Theorem 6. Let δ, αi, βi, γi, δi ∈ C, c = Reδ > 0, M0 the positive solution of the equation
(4), M0 = 1, 5936... and fi ∈ B (µi, λi), gi, hi, ki ∈ A for all z ∈ U, i = 1, n. Suppose also
that

|fi (z)| < Mi,

∣∣∣∣∣g
′′
i (z)

g
′
i(z)

∣∣∣∣∣ < M0,

∣∣∣∣∣h
′′
i (z)

h
′
i(z)

∣∣∣∣∣ < M0,

∣∣∣∣∣k
′′
i (z)

k
′
i(z)

∣∣∣∣∣ < M0,

where Mi are positive real numbers. If

1

c

n∑
i=1

[
|αi − 1| (2− λi)Mµi−1

i + 2 |γi|
]

+
2

(2c+ 1)
2c+1
2c

n∑
i=1

[|βi|M0 + 2 |δi|M0] ≤ 1, (13)

then the function Tn, defined by (3) is in the class S.

Proof. It is easily seen that Tn is regular in U.
Therefore, we get

1− |z|2c

c

∣∣∣∣zT ′′n (z)

T ′n(z)

∣∣∣∣ ≤ 1− |z|2c

c

n∑
i=1

[
|αi − 1|

(∣∣∣∣zf ′i(z)fi(z)

∣∣∣∣+ 1

)
+ |βi|

∣∣∣∣∣zg
′′
i (z)

gi(z)

∣∣∣∣∣
]

+

+
1− |z|2c

c

n∑
i=1

[
|γi|
(∣∣∣∣zh′i(z)hi(z)

− 1

∣∣∣∣+

∣∣∣∣zk′i(z)ki(z)
− 1

∣∣∣∣)+ |δi|
(∣∣∣∣zh′′i (z)h′i(z)

∣∣∣∣+

∣∣∣∣zk′′i (z)

k′i(z)

∣∣∣∣)] .
From hypothesis and applying Theorem 3, we have∣∣∣∣∣zh

′
i(z)

hi(z)
− 1

∣∣∣∣∣ < 1,

∣∣∣∣∣zk
′
i(z)

ki(z)
− 1

∣∣∣∣∣ < 1.

Also, applying the General Schwarz Lemma to the functions fi, i = 1, n, we obtain

|fi (z)| ≤Mi |z| .

Thus, we find that

1− |z|2c

c

∣∣∣∣zT ′′n (z)

T ′n(z)

∣∣∣∣ ≤ 1− |z|2c

c

n∑
i=1

|αi − 1|

(∣∣∣∣f ′i (z)( z

fi(z)

)µi∣∣∣∣ ∣∣∣∣fi(z)z

∣∣∣∣µi−1 + 1

)
+



C. Bărbatu, D. Breaz / Eur. J. Pure Appl. Math, 13 (5) (2020), 1285-1299 1294

+
1− |z|2c

c

n∑
i=1

[|βi|M0 |z|+ |γi| (1 + 1) + |δi| (M0 |z|+M0 |z|)] ≤

≤ 1− |z|2c

c

n∑
i=1

[
|αi − 1|

(∣∣∣∣f ′i (z)( z

fi(z)

)µi∣∣∣∣+ 1

)
Mµi−1
i + |βi|M0 |z|

]
+

+
1− |z|2c

c

n∑
i=1

(2 |γi|+ 2 |δi|M0 |z|) ≤

≤ 1− |z|2c

c

n∑
i=1

[
|αi − 1| (2− λi)Mµi−1

i + |βi|M0 |z|+ (2 |γi|+ 2 |δi|M0 |z|)
]
. (14)

Since

max
|z|≤1

(
1− |z|2c

)
|z|

c
=

2

(2c+ 1)
2c+1
2c

, (15)

from (14) and (15), we obtain

1− |z|2c

c

∣∣∣∣zT ′′n (z)

T ′n(z)

∣∣∣∣ ≤ 1

c

n∑
i=1

[
|αi − 1| (2− λi)Mµi−1

i + 2 |γi|
]

+

+
2

(2c+ 1)
2c+1
2c

n∑
i=1

[|βi|M0 + 2 |δi|M0] . (16)

Using (13) from (16), we have

1− |z|2c

c

∣∣∣∣zT ′′n (z)

T ′n(z)

∣∣∣∣ ≤ 1

for all z ∈ U.
By Theorem Pascu it results that Tn ∈ S.

3. Corollaries and consequences

First of all, upon setting δ = 1 and γi = 0 in Theorem 4, we immediately arrive at the
following corollary:

Corollary 1. Let γ, αi, βi, δi ∈ C, c = Reγ > 0 and Mi, Ni, Ri, Si ≥ 1, i = 1, n, such
that

(2c+ 1)
2c+1
2c

n∑
i=1

|αi − 1|
[
1 + (2− λi)Mµi−1

i

]
+

+2c
n∑
i=1

[|βi|Ni + |δi| (Ri + Si)] ≤ c (2c+ 1)
2c+1
2c . (17)



C. Bărbatu, D. Breaz / Eur. J. Pure Appl. Math, 13 (5) (2020), 1285-1299 1295

If fi ∈ B (µi, λi) , gi, hi, ki ∈ A, satisfies

|fi (z)| < Mi,

∣∣∣∣∣g
′′
i (z)

g
′
i(z)

∣∣∣∣∣ ≤ Ni,

∣∣∣∣∣h
′′
i (z)

h
′
i(z)

∣∣∣∣∣ ≤ Ri,
∣∣∣∣∣k
′′
i (z)

k
′
i(z)

∣∣∣∣∣ ≤ Si,
for all z ∈ U, i = 1, n, then the integral operator Yn, defined by

Yn(z) =

∫ z

0

n∏
i=1

[(
fi(t)

t

)αi−1 (
gi(t)

′)βi (hi′(t))
ki
′(t)

)δi]
dt, (18)

is in the class S.

Remark 2. Taking in (18) δi = 0, we obtain Theorem that was obtained in [28].

If we consider δ = 1 and βi = 0 in Theorem 4, obtain the next corollary:

Corollary 2. Let γ, αi, γi, δi ∈ C, c = Reγ > 0 and Mi, Pi, Qi, Ri, Si ≥ 1, i = 1, n, such
that

(2c+ 1)
2c+1
2c

n∑
i=1

{
|αi − 1|

[
1 + (2− λi)Mµi−1

i

]
+ |γi|

[
2 + (2− ηi)P νi−1i

]}
+

+ (2c+ 1)
2c+1
2c

n∑
i=1

|γi| (2− ρi)Qθi−1i + 2c

n∑
i=1

|δi| (Ri + Si) ≤ c (2c+ 1)
2c+1
2c . (19)

If fi ∈ B (µi, λi), hi ∈ B (νi, ηi), ki ∈ B (θi, ρi), satisfies

|fi (z)| < Mi, |hi (z)| < Pi, |ki (z)| < Qi,

∣∣∣∣∣h
′′
i (z)

h
′
i(z)

∣∣∣∣∣ ≤ Ri,
∣∣∣∣∣k
′′
i (z)

k
′
i(z)

∣∣∣∣∣ ≤ Si,
for all z ∈ U, i = 1, n, then the integral operator Xn, defined by

Xn(z) =

∫ z

0

n∏
i=1

[(
fi(t)

t

)αi−1(hi(t)
ki(t)

)γi (hi′(t))
ki
′(t)

)δi]
dt, (20)

is in the class S.

Remark 3. To the integral operator given by (20) if we take αi−1 = 0, we obtain another
known result proven in [25].

If we consider δ = 1 and αi − 1 = 0 in Theorem 4, obtain the next corollary:

Corollary 3. Let γ, βi, γi, δi ∈ C, c = Reγ > 0 and Ni, Pi, Qi, Ri, Si ≥ 1, i = 1, n, such
that

(2c+ 1)
2c+1
2c

n∑
i=1

|γi|
[
2 + (2− ηi)P νi−1i + (2− ρi)Qθi−1i

]
+



C. Bărbatu, D. Breaz / Eur. J. Pure Appl. Math, 13 (5) (2020), 1285-1299 1296

+2c

n∑
i=1

[|βi|Ni + |δi| (Ri + Si)] ≤ c (2c+ 1)
2c+1
2c . (21)

If gi ∈ A, hi ∈ B (νi, ηi), ki ∈ B (θi, ρi), satisfies∣∣∣∣∣g
′′
i (z)

g
′
i(z)

∣∣∣∣∣ ≤ Ni, |hi (z)| < Pi, |ki (z)| < Qi,

∣∣∣∣∣h
′′
i (z)

h
′
i(z)

∣∣∣∣∣ ≤ Ri,
∣∣∣∣∣k
′′
i (z)

k
′
i(z)

∣∣∣∣∣ ≤ Si,
for all z ∈ U, i = 1, n, then the integral operator Dn, defined by

Dn(z) =

∫ z

0

n∏
i=1

[(
gi(t)

′)βi (hi(t)
ki(t)

)γi (hi′(t))
ki
′(t)

)δi]
dt, (22)

is in the class S.

Remark 4. If in (22) we put βi = 0, than we obtain Theorem that was obtained in [25].

If we consider δ = 1 and δi = 0 in Theorem 4, obtain the next corollary:

Corollary 4. Let γ, αi, βi, γi ∈ C, c = Reγ > 0 and Mi, Ni, Pi, Qi ≥ 1, i = 1, n, such
that

(2c+ 1)
2c+1
2c

n∑
i=1

{
|αi − 1|

[
1 + (2− λi)Mµi−1

i

]
+ |γi|

[
2 + (2− ηi)P νi−1i

]}
+

+ (2c+ 1)
2c+1
2c

n∑
i=1

|γi| (2− ρi)Qθi−1i + 2c
n∑
i=1

|βi|Ni ≤ c (2c+ 1)
2c+1
2c . (23)

If fi ∈ B (µi, λi) , gi ∈ A, hi ∈ B (νi, ηi), ki ∈ B (θi, ρi), satisfies

|fi (z)| < Mi,

∣∣∣∣∣g
′′
i (z)

g
′
i(z)

∣∣∣∣∣ ≤ Ni, |hi (z)| < Pi, |ki (z)| < Qi,

for all z ∈ U, i = 1, n, then the integral operator Sn, defined by

Sn(z) =

∫ z

0

n∏
i=1

[(
fi(t)

t

)αi−1 (
gi(t)

′)βi (hi(t)
ki(t)

)γi]
dt, (24)

is in the class S.

Remark 5. Taking in (24) γi = 0, we obtain a known result proven in [28].

Letting µi = νi = θi = Mi = Ni = Pi = Qi = Ri = Si = 1 and ρi = ηi = λi for all
i = 1, n in Theorem 4, we have:



REFERENCES 1297

Corollary 5. Let δ, γ, αi, βi, γi, δi ∈ C, c = Reγ > 0 and 0 ≤ λi < 1, i = 1, n, such that

(2c+ 1)
2c+1
2c

n∑
i=1

(3− λi) (|αi − 1|+ 2 |γi|) + 2c
n∑
i=1

(|βi|+ 2 |δi|) ≤ c (2c+ 1)
2c+1
2c . (25)

If gi ∈ A, fi,hi, ki ∈ S∗ (λi) and

|fi (z)| < 1,

∣∣∣∣∣g
′′
i (z)

g
′
i(z)

∣∣∣∣∣ ≤ 1, |hi (z)| < 1, |ki (z)| < 1,

∣∣∣∣∣h
′′
i (z)

h
′
i(z)

∣∣∣∣∣ ≤ 1,

∣∣∣∣∣k
′′
i (z)

k
′
i(z)

∣∣∣∣∣ ≤ 1,

for all z ∈ U, i = 1, n, then for every δ, Reδ ≥ Reγ, the function Tn, defined by (3) is in
the class S.

Letting n = 1, δ = γ and αi − 1 = βi = γi in Theorem 5, we obtain:

Corollary 6. Let c, δ ∈ C with Reδ > 0 and M,N,P,Q,R, S ≥ 1. Suppose that f ∈
B (µ, λ), g ∈ A, h ∈ B (ν, η), k ∈ B (θ, ρ), such that

|f (z)| < M,

∣∣∣∣∣zg
′′
(z)

g′(z)

∣∣∣∣∣ < N, |h (z)| < P, |k (z)| < Q,

∣∣∣∣∣zh
′′
(z)

h′(z)

∣∣∣∣∣ < R,

∣∣∣∣∣zk
′′
(z)

k′(z)

∣∣∣∣∣ < S,

for all z ∈ U. If

Reδ ≥ |δ|
[
(2− λ)Mµ−1 + (2− η)P ν−1 + (2− ρ)Qθ−1 +N +R+ S + 3

]
(26)

and

|c| ≤ 1− |δ|
Reδ

[
(2− λ)Mµ−1 + (2− η)P ν−1 + (2− ρ)Qθ−1 +N +R+ S + 3

]
, (27)

then the integral operator T , defined by

T (z) =

[
α

∫ z

0
tα−1

(
f(t)g′(t)

h(t)

k(t)

h′(t))

k′(t)

)α−1
dt

] 1
α

, (28)

is analytic and univalent in U.

References

[1] C Brbatu, D Breaz. Classes of an univalent integral operator. Studia Univ.”Babes-
Bolyai”, Cluj-Napoca, Mathematica, to appear.

[2] J Becker. Lëownersche Differentialgleichung und quasikonform fortsezbare schlichte
Funktionen. J Reine Angew. Math., 255: 23-43, 1972.

[3] D Breaz, N Breaz. Two Integral Operators. Studia Univ.”Babes-Bolyai”, Cluj-Napoca,
Mathematica, 47(3): 13-21, 2002.



REFERENCES 1298

[4] D Breaz, S Owa, N Breaz. A new integral univalent operator. Acta Universitatis
Apulensis, Alba-Iulia, 16: 11-16, 2008.

[5] R Bucur, D Breaz. Univalence conditions for a new general integral operator. Int.
Elect. J. of Pure and Applied Math., 9(3): 215-223, 2015.

[6] R Bucur, D Breaz. Properties of a general integral operator. Advances in Mathematics:
Scientific Journal, 5(1): 5764, 2016.

[7] R Bucur, D Breaz. Univalence Conditions and Properties of a New General Integral
Operator. Carpathian J. Math., 32(2): 157 - 164, 2016.

[8] R Bucur, L Andrei, D Breaz. Univalence criterion, starlikeness and convexity for a
new integral operator. Proceedings of the ICTAMI, Alba Iulia, 17-26, 2015.

[9] R Bucur, L Andrei, D Breaz. Geometric Properties of a New Integral Operators.
Abstract and Applied Analysis, Article Id 430197: 6 pg., 2015.

[10] R Bucur, L Andrei, D Breaz. Some Results of a New Integral Operator. Journal of
Computational Analysis and Applications, 21(6): 1017-1023 2016.

[11] R Bucur, L Andrei, D Breaz. Properties of a New Integral Operator. Analele Univer-
sitatii Ovidius, Constanta, 24(2): 127-136, 2016.

[12] B A Frasin. Order of convexity and univalence of general integral operator. Journal
of the Franklin, 348: 1012-1019, 2011.

[13] B A Frasin, M Darus. On certain analytic univalent functions. Internat. J. Math. and
Math. Sci., 25(5): 305-310, 2001.

[14] B A Frasin, J Jahangiri. A new and comprehensive class of analytic functions. Analele
Univ. Oradea, Fasc. Math., XV: 59-62, 2008.

[15] I J Kim, E P Merkes. On an integral of powers of a spirallike function. Kyungpook
Math. J., 12(2): 249-253, 1972.

[16] O Mayer. The Functions Theory of the One Variable Complex . Acad. Ed., Bucuresti,
Romania, 101-117, 1981.

[17] P T Mocanu, I Şerb. A sharp simple criterion for a subclass of starlike functions.
Complex variables, 32:161-168, 1997.

[18] V T Nguyen, A Oprea, D Breaz. Convexity properties for a new integral operator.
Acta Universitatis Apulensis, 51: 75-87, 2017.

[19] A Oprea, D Breaz, H M Srivastava. Univalence conditions for a new family of integral
Operators. Filomat, Serbia, 30:(5): 1243-1251, 2016.



REFERENCES 1299

[20] H Oversea. Integral operators of Bazilvic type. Bull. Math. Bucuresti, 37: 115-125,
1993.

[21] N N Pascu. An a univalence criterion II. Itinerant Seminar on Functional Equations,
Approximation and Convexity, Cluj Napoca, 153-154, 1985.

[22] N N Pascu. An improvement of Becker’s univalence criterion of univalence. Proceed-
ings of the Commemorative Session Simion Stoilov, Braşov, 1987.

[23] N N Pascu, V Pescar. On the integral operators Kim-Merkes and Pfaltzgraff. Math-
ematica, Univ. Babes-Bolyai, Cluj-Napoca, 32(55)2: 85-192, 1990.

[24] V Pescar. A new generalization of Ahlfors’s and Becker’s criterion of univalence. Bull.
Malaysian Math. Soc., 19(2): 53-54, 1996.

[25] V Pescar. New univalence criteria for some integral operators. Studia Univ. Babes-
Bolyai Math., 59(2): 185-192, 2014.

[26] V Pescar, S Owa. Univalence of certain integral operators. Int. J. Math. Math. Sci.,
23: 697-701, 2000.

[27] J Pfaltzgraff. Univalence of the integral of (f ′(z))λ. Bull. London Math. Soc., 7(3):
54-256, 1975.

[28] N Ularu. Convexity properties for an integral operator . Acta Univ. Apulensis Math.
Inform., 27: 115-120, 2011.