Hadamard product of analytic functions and some special regions and curves
© Piejko and Sokół; licensee Springer. 2013
Received: 20 April 2013
Accepted: 15 August 2013
Published: 3 September 2013
In this paper we present some new applications of convolution and subordination in geometric function theory. The paper deals with several ideas and techniques used in this topic. Besides being an application to those results, it provides interesting corollaries concerning special functions, regions and curves.
Dedicated to Professor Hari M Srivastava
The convolution has the algebraic properties of ordinary multiplication. We now look at some problems on convolution and at some of the relations between the convolution and the subordination. One can consider the following problems.
If and , where are univalent in Δ, then, by (1.2), Problem 2 above becomes the following one:
Ruscheweyh and Sheil-Small  proved the Pòlya-Schoenberg conjecture that the class of convex univalent functions is preserved under convolution, namely . They proved also that the class of starlike functions and the class of close-to-convex functions are closed under convolution with the class . Another solution of Problem 1 is the following theorem.
Theorem A (see )
Problem 2, when the sets M and K are discs with center and radius , was considered in  and in other papers. One of results obtained there is the following theorem.
Theorem B (see )
In this paper we shall look for a solution of the following modified Problem 3.
The next theorem is a solution of Problem 4.
Theorem C (see )
is the Libera operator .
Theorem C improves an earlier result from  with the stronger assumption and the same conclusion. Moreover, if we assume more, that and f is univalent in Δ, then in (1.3) instead of we can put any convex univalent function. This is contained in the following result due to Ruscheweyh and Stankiewicz .
Theorem D (see )
This relationship allows us to obtain several subordination results about convolution. Note that there are no assumptions about the normalization of functions. It is one of the solutions of Problem 3. Many of the convolution properties were studied by Ruscheweyh in , and they have found many applications in various fields. The book  is also an excellent survey of the results. For the recent results on the Hadamard product in geometric function theory, see [9–13].
2 A family of functions
where . In Section 3 we prove that for given the curve , is the Booth lemniscate. Let denote a class of starlike functions consisting of functions f such that .
Lemma 2.1 The function given by (2.1) is a starlike univalent function for .
Thus, it is obvious that is a starlike univalent function for . □
Lemma 2.2 Let the function be of the form (2.1), and let be given by (1.4). If , then .
which is true for . □
Moreover, analogous consideration as in the above proof leads us to the fact that for . In Lemma 3.2 we show that for .
is the Libera operator.
Therefore, using Theorem C, we get (2.5). □
3 A family of sets
- 1.Suppose that . Let be a circle with the center S and the length of the radius R such that
- 2.Let . Then the curve (3.7) consists of points M such that
We say that a closed curve γ is convex when it is boundary of a convex bounded domain. Otherwise, we say that the curve γ is concave.
is attained at one point only. If , then the curve , is concave and (3.10) is attained twice. Moreover, in both cases this curve is symmetric with respect to both axes.
Corollary 3.3 If , then .
By Lemma 2.1 the functions , , are starlike univalent, hence by the geometric interpretation (1.2) of a subordination under univalent functions, we obtain from Theorem 2.3 the following corollary.
and hence by Lemma 3.1 the subordination (3.16) is equivalent to (3.13). □
The function is univalent as the convolution of a convex univalent function, so by (1.2) we obtain (3.18). □
Hence by (1.2) we obtain (3.21). □
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