Abstract
In this article, we determine certain conditions under which the partial sums of the multiplier integral operators of analytic univalent functions of bounded turning are also of bounded turning.
multiplier integral operator; partial sums; bounded turning
Partial sums of analytic functions of bounded turning with applications
Maslina Darus* * Corresponding author. ; Rabha W. Ibrahim
School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia, Bangi 43600, Selangor Darul Ehsan, Malaysia E-mail: maslina@ukm.my / rabhaibrahim@yahoo.com
ABSTRACT
In this article, we determine certain conditions under which the partial sums of the multiplier integral operators of analytic univalent functions of bounded turning are also of bounded turning.
Mathematical subject classification: 30C45.
Key words: multiplier integral operator, partial sums, bounded turning.
1 Introduction
Let be the class of functions analytic in the open unit disk U = {z : z ∈ and |z| < 1} and [a, n] be the subclass of consisting of functions of the form
(z) = a + anzn + an+1zn+1 + ··· .
Let be the subclass of consisting of functions of the form
For 0
µ < 1, let B(µ) denote the class of functions of the form (1) so that ℜ {'} > µ ∈ U. The functions in B(µ) are called functions of bounded turning (c.f. [1, Vol. II]). By the Nashiro-Warschowski Theorem (see e.g. [1, Vol. I]) the functions in B(µ) are univalent and also close-to-convex in U.For of the form (1), several interesting families of integral operators, which have been investigated rather extensively in analytic function theory, including each of the following integral operators (see [2-10]),
and
Also, we define a general integral operator as the following:
where
Remark 1.1. When λ = 0, operator (4) gives Noor integral operator (see [11, 12]).
The m-th partial sums of the operators (2-4) are respectively given by
and
It was shown that for a normalized univalent function of the form (1) the partial sums of the Libera integral operator of functions is starlike in |z| < . The number is sharp ([13]). In [14], it was also shown that the partial sums of the Libera integral operator of functions of bounded turning are also of bounded turning. We determine conditions under which the partial sums (5-7) of the multiplier integral operators (2-4) of analytic univalent functions of bounded turning are also of bounded turning. In the sequel we need to the following results.
Lemma 1.1 [14]. For z ∈ U we have
Lemma 1.2 [1, Vol. I]. Let P(z) be analytic in U, such that P(0) = 1, and ℜ (P(z)) > in U. For functions Q analytic in U the convolution function P * Q takes values in the convex hull of the image on U under Q.
The operator (*) stands for the Hadamard product or convolution of two power series in,
2 Main Results
By making use of Lemma 1.1 and Lemma 1.2, we illustrate the conditions under which the m-th partial sums (5-7) of the multiplier integral operators (2-4) of analytic univalent functions of bounded turning are also of bounded turning.
Theorem 2.1. Let ∈ . If < µ < 1 and (z) ∈ B(µ) , then
Proof. Let be of the form (1) and (z) ∈ B(µ) that is
This implies
Now for < µ < 1 we have
then
Applying the convolution properties of power series to (z) we may write
In virtue of Lemma 1.1 and for j = m – 1, we receive
Thus for 0 < a 1 and –1 < b 1 yields
Hence
A computation gives
On the other hand, the power series
satisfies: P(0) = 1 and
Therefore, by Lemma 1.2, we have
This completes the proof of Theorem 2.1. □
In the next corollary, we establish the conditions of the partial sums of the operator (3) to be of bounded turning when is of bounded turning.
Corollary 2.1. Let ∈ . If < µ < 1 and (z) ∈ B(µ) , then Jm(z) ∈
Proof. Setting a = 1 and b = c in Theorem 2.1 leads to Corollary 2.1.
Corollary 2.2. Let ∈ . If < µ < 1 and (z) ∈ B(µ) , then Lm(z) ∈ , where L(z) denotes the Libera integral operator:
and its m-th partial sums are given by
Proof. Setting a = b = 1 in Theorem 2.1 leads to Corollary 2.2.
Corollary 2.3. Let ∈ . If < µ < 1 and (z) ∈ B(µ) , then Sm(z) ∈ , where Sk(z) denotes the integral operator which analogous to one defined by Sălăgean (see [15]):
and its m-th partial sums are given by
Proof. Setting a = k, b = 0 in Theorem 2.1 leads to Corollary 2.3.
Theorem 2.2. Let ∈ . If < µ < 1 and (z) ∈ B(µ) , then
Proof. By the hypotheses of the theorem we have
This implies, for < µ < 1,
then
Applying the convolution properties of power series to (z) we have
In view of Lemma 1.1 with
yields
Hence
Under the conditions given in (16) we obtain
On the other hand, the power series
satisfies: P(0) = 1 and
Therefore, by Lemma 1.2, we have
The proof of Theorem 2.2 is complete.
Corollary 2.4. Let ∈ . If < µ < 1 and (z) ∈ B(µ) , then Sm(z) ∈ , where S(z) defined in (13) of order one.
Proof. Setting λ = 1 in Theorem 2.2 leads to Corollary 2.4.
Acknowledgement. The authors were supported in part by ScienceFund: 04-01-02-SF0425, MOSTI, Malaysia.
Received: 15/III/09.
Accepted: 15/III/09
#CAM-79/09.
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Publication Dates
-
Publication in this collection
19 Mar 2010 -
Date of issue
2010
History
-
Accepted
15 Mar 2009 -
Received
15 Mar 2009