ABSTRACT

In the present article, certain classes of generalized p-valent Robertson functions are considered. Mapping properties of these classes are investigated under certain p-valent integral operators introduced by Frasin recently.

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.

REFERENCES

NOTES

^*Corresponding author.