Partial sums of analytic functions of bounded turning with applications

Partial sums of analytic functions of bounded turning with applications

Maslina Darus^*; Rabha W. Ibrahim

School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia, Bangi 43600, Selangor Darul Ehsan, Malaysia E-mail: [e-mail] / [e-mail]

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

ƒ (z) = a + a_nzⁿ + a_n+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 ƒ of the form (1) so that ℜ {ƒ'} > 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 ƒ 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 ƒ 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 ƒ ∈ . If < µ < 1 and ƒ(z) ∈ B(µ) , then

Proof. Let ƒ be of the form (1) and ƒ(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 ƒ is of bounded turning.

Corollary 2.1. Let ƒ ∈ . If < µ < 1 and ƒ(z) ∈ B(µ) , then J_m(z) ∈

Proof. Setting a = 1 and b = c in Theorem 2.1 leads to Corollary 2.1.

Corollary 2.2. Let ƒ ∈ . If < µ < 1 and ƒ(z) ∈ B(µ) , then L_m(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 ƒ ∈ . If < µ < 1 and ƒ(z) ∈ B(µ) , then S_m(z) ∈ , where S^k(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 ƒ ∈ . If < µ < 1 and ƒ(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 ƒ ∈ . If < µ < 1 and ƒ(z) ∈ B(µ) , then S_m(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.

• [1] A.W. Goodman, Univalent Functions, Vols. I and II, Polygonal Publishing House, Washington, New Jersey (1983).
• [2] S. Owa and H.M. Srivastava, Some applications of the generalized Libera integral operator Proc. Japan Acad. Ser. A Math. Sc., 62 (1986), 125-128.
• [3] H.M. Srivastava and S. Owa, New characterizations of certain starlike and convex generahzed hypergeometric functions J. Nat. Acad Math. India, 3 (1985), 198-202.
• [4] H.M. Srivastava and S. Owa, A certain one-parameter additive family of operators defined on analytic functions J. Math. Anal. Appl., 118 (1986), 80-87.
• [5] J -L. Liu, On a class of p-valent analytic functions Chinese Quart. J. Math., 15(4) (2000), 27-32.
• [6] J.-L. Liu, Some applications of certain integral operator Kyungpook Math. J., 43 (2003), 211-219.
• [7] J.-L Liu, Notes on Jung-Kim-Snvastava integral operator J. Math. Anal. Appl., 294 (2004), 96-103.
• [8] B.A. Uralegaddi and C. Somanatha, Certain integral operators for starlike functions J. Math Res. Exposition, 14-16 (1995).
• [9] J-L Liu, Some properties of two integral operators Soochow J. Math., 25 (1999), 91-96.
• [10] S. Owa, Properties of certain integral operators Georgian Math. J., 2 (1995), 535-545.
• [11] K.I. Noor, On new classes of integral operators J. Nat. Geomet., 16 (1999), 71-80.
• [12] K.I. Noor and M.A. Noor, On integral operators J. Math. Anal. Appl., 238 (1999), 341-352.
• [13] J.L. Liu and S. Owa, On partial sums of the Libera integral operator J. Math. Anal. Appl., 213 (1997), 444-454.
• [14] J.M. Jahangiri, K. Farahmand, Partial sums of functions of bounded turning Inter. J. Math. Math. Sci., 2004(1), (2004), 45-47.
• [15] G.S. Salagean, Subclasses of univalent functions Lecture Notes in Math., 1013 (1983), 362-372. Springer-Verlag, Berlin.

*

Corresponding author.

Publication Dates

• Publication in this collection
  19 Mar 2010
• Date of issue
  2010

History

• Accepted
  15 Mar 2009
• Received
  15 Mar 2009