Mapping Properties of Generalized Robertson Functions under Certain Integral Operators ()
1. Introduction
Let 
be the class of functions 
of the form
which are analytic in the open unit disc
. We write
. A function 
is said to be spiral-like if there exists a real number 
such that
The class of all spiral-like functions was introduced by L. Spacek [1] in 1933 and we denote it by
. Later in 1969, Robertson [2] considered the class 
of analytic functions in 
for which
.
Let 
be the class of functions 
analytic in 
with 
and
where
, 
and 
is real with
.
For
, 
, this class was introduced in [3] and for
, see [4]. For
, 
and
, the class 
reduces to the class 
of functions 
analytic in 
with 
and whose real part is positive.
We define the following classes
For
, 
and
, we obtain the well known classes 
and 
of analytic functions with bounded radius and bounded boundary rotations studied by Tammi [5] and Paatero [6] respectively. For details see [7-12]. Also it can easily be seen that 
and 
Let us consider the integral operators
(1.1)
and
(1.2)
where 
and 
for all 
.
These operators, given by (1.1) and (1.2), are defined by Frasin [13]. If we take
, we obtain the integral operators 
and 
introduced and studied by Breaz and Breaz [14] and Breaz et al. [15], for details see also [16-20]. Also for
, 
in (1.1), we obtain the integral operator studied in [21] given as
and for
, 
, 
in (1.2), we obtain the integral operator
discussed in [22,23].
In this paper, we investigate some propeties of the above integral operators 
and 
for the classes 
and
respectively.
2. Main Result
Theorem 2.1. Let 
for 
with
. Also let 
is real with
, 
,
. If
then 
with
(2.1)
Proof. From (1.1), we have
(2.2)
or, equivalently
(2.3)
Subtracting and adding 
on the right hand side of (2.3), we have
(2.4)
Taking real part of (2.4) and then simple computation gives
(2.5)
where 
is given by (2.1). Since 
for
, we have
(2.6)
Using (2.6) and (2.1) in (2.5), we obtain
Hence 
with 
is given by (2.1).
By setting 
and 
in Theorem 2.1, we obtain the following result proved in [9].
Corollory 2.2. Let 
for 
with
. Also let
,
. If
then 
and 
is given by (2.1).
Now if we take 
and 
in Theorem 2.1, we obtain the following result.
Corollory 2.3. Let 
for 
with
. Also let
,
. If
then 
and 
is given by (2.1).
Letting
, 
, 
and 
in Theorem 2.1, we have.
Corollory 2.4. Let 
with
. Also let
. If
then
with
.
Theorem 2.5. Let 
for 
with
. Also let 
is real is real with
,
,
. If
then 
and 
is given by (2.1).
Proof. From (1.2), we have
or, equivalently
This relation is equivalent to
(2.7)
Taking real part of (2.7) and then simple computation gives us
(2.8)
where 
is given by (2.1). Since 
for
, we have
(2.9)
Using (2.9) in (2.8), we obtain
Hence 
with 
is given by (2.1).
By setting 
and 
in Theorem 2.5, we obtain the following result.
Corollory 2.6. Let 
for 
with
. Also let
,
. If
then 
with 
is given by (2.1).
Letting
, 
, 
and 
in Theorem 2.5, we have.
Corollory 2.7. Let 
with
. Also let
. If
, then
with
.
NOTES