Abstract: The aim of this study was to introduce a certain class of analytic functions containing multiplier transformation in the open unit disk We also investigate some properties of this class.
INTRODUCTION
For
| (1) |
which satisfies the condition
The class of this function is denoted by Pα. For the Hadamard product or convolution of two power series p(z) defined in Eq. 1 and a function q(z) where:
is:
A function
A function
Define an operator as follows:
| (2) |
Clearly, Eq. 2 yields:
Thus, by applying the operator Jcα successively, we can obtain:
Definition 1: The fractional integral of order α is defined, for a function f(z) by (Srivastava and Owa, 1989):
where, the function f(z) is analytic in simply connected region of the complex
z-plane (
Note that (Srivastava and Owa, 1989; Miller and Ross, 1993):
And some of its current properties can be found by Ibrahim and Darus (2008a, b).
Let us define a class
| (3) |
Then in view of Definition 1, we have:
| (4) |
where:
and 0 <
Note that the authors defined and studied different classes (Darus and Ibrahim, 2008; Ibrahim and Darus, 2008c). Thus by applying operator Eq. 2 and 4 yields:
| (5) |
In the present study, we define and study the subclass Nα (m; k ;μ; v) of Pα consisting of p(z) functions which satisfies the inequality:
| (6) |
for some
RESULTS
In this section, we obtain a necessary and sufficient condition and extreme
points for functions p(z)
Theorem 1: Let F(z)εPα defined in Eq. 4 and satisfies the inequality:
| (7) |
where:
Then F(z)εNα (m; k; μ; v) where: 0≤μ<α≤1,
v≥0 and
Proof: Suppose that Eq. 7 is true for
It suffices to show that
| (8) |
Substituting for
Hence the proof.
In virtue of Theorem 1, we now introduce the subclass Nα (m; k; μ; v) which consist of functions F(z) εPα whose coefficients satisfy the inequality Eq. 7. By the coefficient inequality for the class Nα(m; k; μ; v) we see that:
Theorem 2: Let F(z)
Proof: For 0≤v1≤ v2 we receive:
| (9) |
Therefore, if F (z)
Next we determine the extremal points. The determination of the extreme points
of a family T of univalent functions enables us to solve many extremal problems
for
Theorem 3: Let F(z) εPα defined in Eq. 4 and satisfies the inequality (Eq. 7) and
| (10) |
Then F(z)εNα (m; k; μ ;v) where 0≤μ<α≤1,
v≥0 and m
where:
Proof: Suppose that:
Then:
Thus from the definition of the class Nα (m; k; μ; v) we find F (z) ε Nα (m; k; μ; v).
Conversely, suppose that F (z) ε Nα(m; k; μ; v).
Since:
and since βn are arbitrary then we can set:
and:
Then:
This completes the proof of theorem.
Corollary 1: The extreme points of are the functions given by:
where, ψα (m; k; n; μ; v) is defined in Theorem 1 and 1, 2, 3, ..., 0≤μ<α≤1, v≥0.
Now we prove that the condition in Theorem 1 is also necessary for F
| (11) |
Theorem 4: A necessary and sufficient condition for F of the form Eq.
11 to be in Tnα (m; k; μ; v) :=
| (12) |
where, Ψα (m; k; n; μ; v) is defined in Theorem 1.
Proof: In view of Theorem 1, we need only to prove the necessity. If F ∈ Tnα (m; k; μ; v) and z is real then:
or
| (13) |
Substituting for
A computation, we obtain the desired inequality.
Theorem 5: The extreme points of TNα (m; k; μ; v) are the functions given by:
where, ψα(m; k; n; μ; v) is defined in Theorem 1 and 1, 2, 3, ..., 0≤μ<α≤1, v≥0.
CONCLUSION
The class we studied here is the generalization of well-known classes given by Srivastava and Owa (1989). This generalized class ca n be further used to solve many other problems such as the partial differential in complex domain, diffusion equations and Cauchy problems.
ACKNOWLEDGMENT
The work presented here was supported by UKM-ST-06-FRGS0107-2009.