2_dorca.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 6, No. 1, 2013, 11-19

ISSN 1307-5543 – www.ejpam.com

Mapping Properties of Some Classes of Analytic Functions

Under New Generalized Integral Operators

Irina Dorca1,∗, Daniel V. Breaz 2

1 Department of Mathematics, University of Piteşti, Argeş, România
2 Department of Mathematics, University "1st December 1918" of Alba, România

Abstract. In this paper we study the mapping properties with respect to new generalised integral

operator which was studied recently.

2010 Mathematics Subject Classifications: 30C45

Key Words and Phrases: analytic functions, positive coefficients, negative coefficients, integral oper-

ator

1. Introduction

LetH (U) be the set of functions which are regular in the unit disc U ,

A = { f ∈H (U) : f (0) = f ′(0)− 1= 0}

and S = { f ∈A : f is univalent in U}.
In [10] the subfamily T of S consisting of functions f of the form

f (z) = z −
∞
∑

j=2

a jz
j , a j ≥ 0, j = 2,3, . . . , z ∈ U (1)

was introduced.

Thus we have the subfamily S − T consisting of functions f of the form

f (z) = z +

∞
∑

j=2

a jz
j , a j ≥ 0, j = 2,3, . . . , z ∈ U (2)

∗Corresponding author.

Email addresses: irina.dor
a�gmail.
om (I. Dorca), dbreaz�uab.ro (D. Breaz)

http://www.ejpam.com 11 c© 2013 EJPAM All rights reserved.



I. Dorca, D. Breaz / Eur. J. Pure Appl. Math, 6 (2013), 11-19 12

A function f (z) ∈ A is said to be spiral-like if there exists a real number λ, |λ| < π/2,

such that

Re eiλ
z f ′(x)

f (X )
, (z ∈ U).

The class of all spiral-like functions was introduced by L. Spacek [11] and we denote

it by S⋆
λ
. Later, Robertson [9] considered the class Cλ of analytic functions in U for which

z f ′(z) ∈ S⋆
λ
.

Let Pλ
k
(ρ) be the class of functions p(z) analytic in U with p(0) = 1 and

2π
∫

0

�

�

�

�

�

Re eiλp(z)−ρ cosλ

1−ρ

�

�

�

�

�

dθ ≤ kπ cosλ, z = reiθ (3)

where k ≥ 2, 0 ≤ ρ < 1, λ is real with |λ| < π

2
. In case that k = 2, λ = 0, ρ = 0, the class

Pλ
k
(ρ) reduces to the class P of functions p(z) analytic in U with p(0) = 1 and whose real part

is positive.

we recall the well-known classes

Rλk(ρ) =

¨

f (z) : f (z) ∈ A and
z f ′(z)

f (z)
∈ Pλk (ρ), 0≤ ρ < 1

«

,

Vλk (ρ) =

¨

f (z) : f (z) ∈ A and
(z f ′(z))′

f ′(z)
∈ Pλk (ρ), 0≤ ρ < 1

«

.

These classes are introduced and studied in [7].

The purpose of this paper is to develop the mapping properties with respect to a new

generalized integral operator.

2. Preliminary Results

Prof. Breaz [3] has introduced the following integral operators on univalent function

spaces:

J(z) =







β

z
∫

0

�

f ′1(t
n)
�γ1 · . . . ·
h

f ′p(t
n)
iγp

d t







1

β

, (4)

H(z) =







β

z
∫

0

tβ−1
�

f ′1(t)
�γ1 · . . . ·
h

f ′p(t)
iγp

d t







1

β

, (5)

F(z) =

z
∫

0

�

f1(t)

t

�γ1

· . . . ·

�

fp(t)

t

�γp

d t, (6)



I. Dorca, D. Breaz / Eur. J. Pure Appl. Math, 6 (2013), 11-19 13

G(z) =









β

z
∫

0

�

f1(t)

t

�γ1

· . . . ·

�

fp(t)

t

�γp

d t









1

β

, (7)

Fγ,β (z) =







β

z
∫

0

tβ−1

�

f1(t)

t

�
1

γ1

· . . . ·

�

fp(t)

t

� 1

γp

d t







1

β

, (8)

and

Gγ,p(z) =







[p(γ− 1) + 1]

z
∫

0

g
γ−1

1 (t) · . . . · gγ−1
p (t)d t







1

p(γ−1)+1

, (9)

where γi,γ,β ∈ C∀i = 1, p, p ∈ N− {0}, n ∈ N− {0,1}.
Let Dn be the Sălăgean differential operator [see 12] Dn :A →A , n ∈ N, defined as:

D0 f (z) = f (z), D1 f (z) = D f (z) = z f ′(z), Dn f (z) = D(Dn−1 f (z)) (10)

and Dk, Dk :A →A , k ∈ N∪ {0}, of form:

D0 f (z) = f (z), . . . , Dk f (z) = D(Dk−1 f (z)) = z +

∞
∑

n=2

nkanzn. (11)

Definition 1 ([2]). Let β , λ ∈ R, β ≥ 0, λ ≥ 0 and f (z) = z +
∑∞

j=2 a jz
j . We denote by D

β

λ

the linear operator defined by

D
β

λ
: A→ A, D

β

λ
f (z) = z +

∞
∑

j=n+1

[1+ ( j− 1)λ]βa jz
j . (12)

Remark 1. In [1] we have introduced the following operator concerning the functions of form

(1):

D
β

λ
: A→ A, D

β

λ
f (z) = z −

∞
∑

j=n+1

[1+ ( j− 1)λ]βa jz
j . (13)

The neighborhoods concerning the class of functions defined using the operator (13) is studied in

[5].

Remark 2. Let consider the following operator concerning the functions f ∈ S,

S = { f ∈A : f is univalent in U}:

D
n,β

λ1 ,λ2
f (z) = (h ∗ψ1 ∗ f )(z) = z ±

∑

k≥2

[1−λ1(k− 1))]β−1

[1−λ2(k− 1))]β
·

1+ c

k+ c
· C(n, k) · ak · z

k, (14)

where C(n, k) =
(n+1)k−1

(1)k−1
, (·)· is the Pochammer symbol; k ≥ 2, c ≥ 0.



I. Dorca, D. Breaz / Eur. J. Pure Appl. Math, 6 (2013), 11-19 14

The following integral operator is studied in [4], where fi, i = 1 . . . n, n ∈ N, is considered

to be of form (2):

Definition 2. We define the general integral operator Ik,n,λ,µ :An→A by

Ik,n,λ,µ( f1, . . . , fn) = F , (15)

DkF(z) =

z
∫

0

�

Dλ1 f1(t)

t

�µ1

· . . . ·

�

Dλn fn(t)

t

�µn

d t,

where fi ∈A , i ∈ N− {0}, λ= (λ1, . . . ,λn) ∈ N
n
0, µ = (µ1, . . . ,µn) ∈ N

n, n ∈ N and k ∈ N0.

Theorem 1. Let α, γ1, γ2, β ∈ C, Re α = a > 0 and D
n,κ
λ1 ,λ2

f j(z) ∈ A , λ1, λ2, κ ≥ 0, σ ∈ R,

j = 1, p, p ∈ N, D
n,κ
λ1,λ2

f j(z
n) of form (14). If

�

�

�

�

�

(D
n,κ
λ1,λ2

f j(z
n))′′

(D
n,κ
λ1,λ2

f j(z
n))′

�

�

�

�

�

≤
1

n
and

�

�

�

�

�

(D
n,κ
λ1,λ2

f j(z
n))′

(D
n,κ
λ1,λ2

f j(z
n))

�

�

�

�

�

≤
1

n
∀z ∈ U , j = 1, p,

p
∑

j=1

[|δ1
j | · (|2γ1− 1| − |σ|)+ |δ2

j | · (|2γ2− 1| − |σ|)]

|σ · (2γ1− 1) · (2γ2 − 1) · (
p
∏

j=1

δ1
j
· δ2

j
)|

≤ 1,

and

|σ · (2γ1− 1) · (2γ2− 1) · (
p
∏

j=1

δ1
j ·δ

2
j )| ≤

n+ 2a

2
·
�

n+ 2a

n

� 1

n+2a

,

then ∀δ, δ1
j
, δ2

j ∈ C, j = 1 . . . p, Re(β)≥ a, Re(βδ)≥ a, the function

I1(z) =







β

z
∫

0

tβδ−1 ·
p
∏

j=1





((D
n,κ
λ1,λ2

f j(t
n)′)2γ1−1

tσ





δ1
j

·





(D
n,κ
λ1,λ2

f j(t
n))2γ2−1

tσ





δ2
j

d t







1

β

(16)

is univalent for all n ∈ N− {0}.

If we consider the operator D
β

λ
f (z) of form (13) we obtain the following Corollary, whose

proof is similar with the prove of Theorem 1.

Corollary 1. Let α, γ1, γ2, χ ∈ C, Re α = a > 0 and D
β

λ
f j(z) ∈ A , β ≥ 0, λ ≥ 0, σ ∈ R,

D
β

λ
f (zn) of form (13). If

�

�

�

�

�

(D
β

λ
f j(z

n))′′

(D
β

λ
f j(z

n))′

�

�

�

�

�

≤
1

n
and

�

�

�

�

�

(D
β

λ
f j(z

n))′

(D
β

λ
f j(z

n))

�

�

�

�

�

≤
1

n
, ∀z ∈ U , j = 1, p,



I. Dorca, D. Breaz / Eur. J. Pure Appl. Math, 6 (2013), 11-19 15

p
∑

j=1

[|δ1
j
| · (|2γ1− 1| − |σ|)+ |δ2

j | · (|2γ2− 1| − |σ|)]

|σ · (2γ1− 1) · (2γ2− 1) · (
p
∏

j=1

δ1
j
·δ2

j
)|

≤ 1

and

|σ · (2γ1− 1) · (2γ2− 1) · (
p
∏

j=1

δ1
j ·δ

2
j )| ≤

n+ 2a

2
·
�

n+ 2a

n

� 1

n+2a

,

then for all δ, δ1
j , δ

2
j ∈ C, j = 1 . . . p, Re(χ)≥ a, Re(χδ)≥ a, the function

I2(z) =







χ

z
∫

0

tχδ−1

p
∏

j=1





((D
β

λ
f j(t

n)′)2γ1−1

tσ





δ1
j




(D
β

λ
f j(t

n))2γ2−1

tσ





δ2
j

d t







1

χ

(17)

is univalent for ∀n ∈ N− {0}.

Lemma 1 ([6] ). Let u = u1 + iu2, v = v1 + iv2 and Ψ(u, v) be a complex valued function

satisfying the conditions:

(i) Ψ(u, v) is continuous in a domain D ∈ C2, Re

(ii) (1,0) ∈ D and Re Ψ(1,0)> 0,

(iii) Re Ψ(iu2, v1)≤ 0, whenever (iu2, v1) ∈ D and v1 ≤ −
1

2
(1+ u2

2).

If h(z) = 1+
∑

i≥1

ciz
i is an analytic function in U such that (h(z), zh′(z)) ∈ D and

Re Ψ(h(z), zh′(z))> 0 for z ∈ U , then Re h(z)> 0 in U .

Lemma 2 ([8]). Let f (z) ∈ V λ
k
(ρ), 0 ≤ ρ < 1 and λ is real with |λ| < π

2
. Then f (z) ∈ Rλ

k
(β),

where β is one of the root of

2β3 + (1− 2ρ)β2+ (3 sec2λ− 4)β − (1+ 2ρ) tan2λ= 0. (18)

Following we present the mapping properties of the general integral operator of form (16),

giving also several examples which prove its relevance.

3. Main Results

Theorem 2. Let D
n,κ
λ1,λ2

f j(z
n) ∈ Rλ

k
, D

n,κ
λ1,λ2

f j(z
n) of form (14), n ∈ N, λ1, λ2, κ ≥ 0, σ ∈ R,

j = 1, p p ∈ N, for 0≤ ρ < 1. Also let λ be real, |λ|< φ

2
. If

0≤ [ρ − 1]

p
∑

j=1

δa
j + βδ < 1,



I. Dorca, D. Breaz / Eur. J. Pure Appl. Math, 6 (2013), 11-19 16

then I1(z) ∈ V λ
k
(η), I1(z) of form (16), with

η = [ρ− 1]

p
∑

j=1

δa
j + βδ, (19)

β , δ, δa
j
∈ C, a ∈ {1,2}, j = 1, p, Re(βδ)> 0.

Proof. Let consider the notations

h(z) =

z
∫

0

tβδ−1

p
∏

j=1





((D
n,κ
λ1,λ2

f j(t
n)′)2γ1−1

tσ





δ1
j

·





(D
n,κ
λ1,λ2

f j(t
n))2γ2−1

tσ





δ2
j

d t

=

z
∫

0

tβδ−1

p
∏

j=1

h

h1
j (t

n)
iδ1

j
·
h

h2
j (t

n)
iδ2

j
d t

in (16), with α, γ1, γ2, β , δ ∈ C, Re α = a > 0 and D
n,κ
λ1,λ2

f j(z) ∈ A , n ∈ N, λ1, λ2, κ ≥ 0,

σ ∈ R, j = 1, p, p ∈ N.

From Theorem 1, we obtain

[I1(z)]′′

[I1(z)]′
=

�

1

β
− 1

�

·
h′(z)

h(z)
+βδ ·

1

z
+







p
∑

j=1,a∈{1,2}

δa
j ·
[ha

j (z)]
′

ha
j
(z)
−

1

z







which is equivalently to

eiλ

�

1+
z[I1(z)]′′

[I1(z)]′

�

= eiλ·

�
�

1

β
− 1

�

·
zh′(z)

h(z)
+ βδ

�

+eiλ·







p
∑

j=1,a∈{1,2}

δa
j ·

z[ha
j
(z)]′

ha
j
(z)
− 1





+eiλ

(20)

Furthermore, we have

Re

�

eiλ

�

1+
z[I1(z)]′′

[I1(z)]′

��

≤ (βδ− 1)+ Re





e
iλ ·







p
∑

j=1,a∈{1,2}

δa
j ·

z[ha
j
(z)]′

ha
j
(z)
− 1





+ eiλ





 ,

which can be written as following

Re

�

eiλ

�

1+
z[I1(z)]′′

[I1(z)]′

��

≤ Re





e
iλ ·







p
∑

j=1,a∈{1,2}

δa
j ·

z[ha
j
(z)]′

ha
j
(z)
− 1





+ βδeiλ





 .

Subtracting and adding ρ cosλ

p
∑

j=1,a∈{1,2}

δa
j

on the left hand side of (20) and then taking

the real part, we have

Re

�

eiλ

�

1+
z[I1(z)]′′

[I1(z)]′

�

−η cosλ

�

≤
p
∑

j=1,a∈{1,2}

δa
j Re



eiλ ·
[ha

j (z)]
′

ha
j
(z)
−ρ cosλ



 , (21)



I. Dorca, D. Breaz / Eur. J. Pure Appl. Math, 6 (2013), 11-19 17

where η is given by (19).

Integrating (21) and then using (19), we have

∫ 2π

0

�

�

�

�

�

Re

�

eiλ

�

1+
z[I1(z)]′′

[I1(z)]′

�

−η cosλ

�

�

�

�

�

�

dθ

≤
1−η

1−ρ

∫ 2π

0

�

�

�

�

�

Re



eiλ ·
[ha

j
(z)]′

ha
j
(z)
−ρ cosλ





�

�

�

�

�

dθ . (22)

Since f j(z
n) ∈ Rλ

k
(ρ), j = 1, p, p, n ∈ N− {0}, we obtain

∫ 2π

0

�

�

�

�

�

Re



eiλ ·
[ha

j
(z)]′

ha
j
(z)
−ρ cosλ





�

�

�

�

�

dθ ≤ (1−ρ)kπ cosλ. (23)

Using (22) and (23), we have

∫ 2π

0

�

�

�

�

�

Re

�

eiλ

�

1+
z[I1(z)]′′

[I1(z)]′

�

−η cosλ

�

�

�

�

�

�

dθ ≤ (1−η)kπ cosλ.

Hence I1(z) ∈ V λ
k
(η) with η given by (19).

Remark 3. If we consider the operator D
β

λ
f (z) ∈ Rλ

k
(ρ) of form (13) we obtain similar result as

in Theorem 2.

Remark 4. If we apply the operator (10) to the integral operator F(z) of form (6), we obtain

the result from [8].

Next we give few examples of particular cases which can be found in literature.

Let β = 0 in D
β

λ
f (z) of form (12) or (13). So we have that D0

λ
f (z) = f (z),∀λ ≥ 0. We

will use this form of the integral operator, where the function f is of form (2) with respect to

the operator (17). For further simplification, we consider that γ1 = γ2 = 1, and δ = 1 (except

of Example 4).

For the first four examples we consider δ1
j
= 0, j = 1, p, p ∈ N−{0}, n= 1.

Example 1. If σ = 1, χ = 1 and we use the notation δ2
j = γ j, j = 1, p, p ∈ N−{0}, we obtain the

operator F(z) of form (6). F(z) ∈ V λ
k
(η) if 0≤ (ρ−1)

p
∑

j=1

γ j+1< 1 with η = (ρ−1)
p
∑

j=1

γ j+1.

Example 2. If σ = 1 we obtain the operator G(z) of form (7) for δ2
j
= γ j , j = 1, p, p ∈ N−{0}.

G(z) ∈ V λ
k
(η) if 0≤ (ρ− 1)

p
∑

j=1

γ j + 1< 1 with η= (ρ− 1)
p
∑

j=1

γ j + 1.



REFERENCES 18

Example 3. If σ = 1 and we use the notation δ2
j
= 1/γ j, j = 1, p, p ∈ N− {0}, we obtain the

operator Fγ,β (z) of form (8). Fγ,β(z) ∈ Vλ
k
(η) if 0≤ (ρ− 1)

p
∑

j=1

1

γ j
+ β < 1 with

η = (ρ− 1)
p
∑

j=1

γ j + β .

Example 4. If σ = 0 we obtain the operator Gγ,p(z) of form (9) for χ = [p(γ− 1) + 1], δ = 1

χ

and δ2
j = γ− 1, Gγ,p(z) ∈ Vλ

k
(η) if 0≤ (1−ρ)

p
∑

j=1

γ j + 1< 1 with η = (ρ− 1)
p
∑

j=1

γ j + 1.

For the next two examples we consider δ2
j = 0, j = 1, p, p ∈ N− {0}, and σ = 0.

Example 5. a) If χ = 1, δ = 1, we obtain a particular case of the function J(z) of form

(4), in which β = 1,∀n ∈ N − {0}. J(z) ∈ Vλ
k
(η) if 0 ≤ (1− ρ)

p
∑

j=1

γ j + 1 < 1 with

η = (ρ− 1)
p
∑

j=1

γ j + 1.

b) If δ = 1

χ
, δ1

j
= γ j , j = 1, p, p ∈ N−{0}, we obtain the operator J(z) of form (4), in which

β = 1,∀n ∈ N−{0}. J(z) ∈ Vλ
k
(η) if 0≤ (1−ρ)

p
∑

j=1

γ j+1< 1 with η = (ρ−1)
p
∑

j=1

γ j+1.

Example 6. If n = 1, δ = 1

χ
, we obtain the operator H(z) of form (5) for δ1

j
= γ j , j = 1, p,

p ∈ N− {0}. F(z) ∈ V λ
k
(η) if 0≤ (1−ρ)

p
∑

j=1

γ j + β < 1 with η = (ρ− 1)
p
∑

j=1

γ j + β .

ACKNOWLEDGEMENTS This work was partially supported by the strategic project POSDRU

107/1.5/S/77265, inside POSDRU Romania 2007-2013 co-financed by the European Social

Fund-Investing in People. The authors thank the readers of European Journal of Pure and

Applied Mathematics, for making our journal successful.

References

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Daniel v. Breaz, editor, Proceedings of the Interational Coference on Theory and Applica-

tions of Mathematics and Informatics., pages 101–112, Alba Iulia, 2011. ICTAMI.

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