Abstract
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.
MSC:30C45, 30C80.
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Dedication
Dedicated to Professor Hari M Srivastava
1 Introduction
By ℋ we denote a class of analytic functions in the unit disc . Let denote the subclass of ℋ consisting of functions normalized by , and satisfying
Of course the functions from map the unit disc onto convex domains.
We say that the function is subordinate to the function in the unit disc Δ (written ) if and only if there exists an analytic function such that , and for . Therefore in Δ implies . If, additionally, g is univalent in Δ, then
Recall that the Hadamard product or convolution of two power series,
is defined as
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.
Problem 1 Find the subclasses , ℬ,
of the class ℋ such that
where
Problem 2 Let the sets and be given. Find a set , as small as possible, such that
where is some subclass of ℋ.
If and , where are univalent in Δ, then, by (1.2), Problem 2 above becomes the following one:
Problem 3 Let the univalent functions F, G be given. Find a function H, , and a class such that
Ruscheweyh and Sheil-Small [1] 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 [2])
Let us denote
Suppose that , . Then
where .
Problem 2, when the sets M and K are discs with center and radius , was considered in [3] and in other papers. One of results obtained there is the following theorem.
Theorem B (see [3])
Let , ; . If and , , then the following implication holds:
where
In this paper we shall look for a solution of the following modified Problem 3.
Problem 4 Find a function and a class such that
The next theorem is a solution of Problem 4.
Theorem C (see [4])
Assume that , , and that it satisfies
If , then
where
is the Libera operator [5].
If , then we have
Therefore, Theorem C may be written as
where
Theorem C improves an earlier result from [6] 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 [7].
Theorem D (see [7])
Let F and G be convex univalent in Δ. Then, for all functions ,
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 [8], and they have found many applications in various fields. The book [8] 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
Throughout this section we consider the family of analytic 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 .
Proof We have
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 .
Proof Assume that , , and set
By (1.4) if for all , , then for all . After some calculations, we obtain
where . Thus we shall find in such that
It is easy to verify that for all and for all , and that for all . Hence, to find in satisfying (2.4), we may restrict our considerations to the interval . Then, analyzing as a function of one variable x, we get
for all . Moreover, we see that
Therefore, if , then
To find in satisfying (2.4), it is sufficient to solve the inequality
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 .
Theorem 2.3 If and , then
where
is the Libera operator.
Proof By Lemma 2.2 we obtain
Therefore, using Theorem C, we get (2.5). □
Corollary 2.4 If and
then
It is known [14] that if
then
Hence, for , Corollary 2.4 gives also that
because for , , ,
Theorem 2.5 If , , and the functions and are in the class ℋ, then
Proof By Theorem 2.3 we have and . By Lemma 2.1 , , so the functions , are in because . Using Theorem D, we obtain
but because
□
3 A family of sets
In this section we investigate the properties of a one-parameter family of the sets , , such that
when and
Lemma 3.1 Let , then , where
Proof The function is analytic in the unit disc, so for the proof we need to find an image of the circle under the function . For and for , we have
Let us define
and
Then, after some calculations, we can obtain from (3.4) and (3.5) that
Therefore, using the relations (3.3)-(3.5), we can find that is an algebraic curve of order four whose equation in orthogonal Cartesian coordinates is
Thus, because for , the set is bounded by the curve (3.7). Moreover, for , the function becomes
and it is easy to see that , see Figure 1. Then the proof is completed. □
Notice that a curve described by
is called the Booth lemniscate, named after Booth [15, 16]. The Booth lemniscate is called elliptic if , while for it is termed hyperbolic. Thus it is clear that the curve (3.7) is the Booth lemniscate of elliptic type. The Booth lemniscate is a special case of a Persian curve. The Booth lemniscate of elliptic type can be described geometrically in the following equivalent ways.
-
1.
Suppose that . Let be a circle with the center S and the length of the radius R such that
A ray is drawn from the point x on the circle through the origin o that cuts also the circle at the point z. A point v is also on the ray and satisfies
If the point x goes along the circle , then the point v describes the curve (3.7), see Figure 2.
-
2.
Let . Then the curve (3.7) consists of points M such that
where and the points , are the focuses
see Figure 3.
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.
Lemma 3.2 Suppose that is given by (2.1). If , then the curve , is convex and
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.
Proof If , then the curve , becomes a circle and it is clear that (3.10) is attained one time only. Suppose, in the sequel, that . From (2.2) and (2.3) we have
The denominator is positive, hence for the convexity of we need the nominator to be positive. Since for , then the nominator is positive because
From (3.5) we have
Then if and only if . That is possible when because . Therefore, using the elementary considerations, we can find that if , then the curve , is concave and (3.10) is attained twice, when . In this case,
see Figure 3. Moreover, if , then (3.10) is attained at one point only such that
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.
Corollary 3.4 If and , then
Theorem 3.5 Assume that , , and that the functions and are in the class ℋ with . If
then
Proof Using Lemma 3.1, we may rewrite (3.12) as
By Corollary 3.3, is convex univalent in Δ for , so , are convex univalent in Δ. From (1.2) and from (3.14), we obtain
By Theorem D, we get from (3.15)
But
and hence by Lemma 3.1 the subordination (3.16) is equivalent to (3.13). □
Theorem 3.6 Assume that and that the functions and are in the class ℋ with . Let be the disc with center and radius R with . If
then
Proof Using Lemma 3.1, we may rewrite (3.17) as
where
maps Δ onto . By Corollary 3.3, is convex univalent in Δ, also is convex univalent in Δ. Therefore, by Theorem D we obtain
The function is univalent as the convolution of a convex univalent function, so by (1.2) we obtain (3.18). □
Theorem 3.7 Assume that and that the functions and are in the class ℋ with . Let , , , be the half-plane. If
then
Proof Using Lemma 3.1, we may rewrite (3.20) as
where
maps Δ onto . By Corollary 3.3, is convex univalent in Δ, also is convex univalent in Δ. Therefore, by Theorem D we obtain
Hence by (1.2) we obtain (3.21). □
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Piejko, K., Sokół, J. Hadamard product of analytic functions and some special regions and curves. J Inequal Appl 2013, 420 (2013). https://doi.org/10.1186/1029-242X-2013-420
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DOI: https://doi.org/10.1186/1029-242X-2013-420