Univalency of convolutions of harmonic mappings
Introduction
Let be the unit disk. We consider the family of complex-valued harmonic functions defined in , where u and v are real harmonic in . Such functions can be expressed as , whereare analytic in . By the Lewy theorem, a harmonic function is locally univalent and sense-preserving if and only if its dilatation satisfies for , see e.g. [5].
For functions and , analytic in , their convolution (or Hadamard product) is defined as
If we take two harmonic functions , we can define in the natural way as where . Convolutions of two harmonic functions were studied in [2], [3], [4], [6], [11].
Let be the class of complex-valued harmonic sense-preserving univalent functions f in , normalized so that , , and .
It is known that if maps the unit disk onto the right half-plane , then it must satisfy the following conditionLet denote the class of harmonic mappings that satisfy (1). They are the so called vertical shears of the conformal half-plane mapping . It was proved in [3] that if , and is locally univalent and sense-preserving, then and is convex in the direction of the real axis. As observed in [4] the assumption of the local univalency of the convolution function in this statement cannot be omitted. However, in some cases these harmonic convolutions are locally univalent. It was shown in [4] that if with dilatation and with dilatation , then the convolution is locally univalent for all if and only if .
In this paper we prove the following. Theorem If , and with , then the function is convex in the direction of the real axis. In particular, if , then is convex.
We will use the following characterization of the class of analytic functions mapping conformally onto a domain convex in one direction due to Royster and Ziegler [10]. Theorem A A nonconstant and analytic function F maps univalently onto a domain convex in the direction of the imaginary axis if and only if there are numbers and , such that
In the proof of our theorem we will also take advantage of the theory of harmonic Hardy spaces. A function harmonic in the unit disk is said to be of class if the integral meansare bounded for ; is simply the collection of bounded harmonic functions on . The usual Hardy spaces consist of the functions in that are analytic in . It is clear that an analytic function belongs to if and only if its real and imaginary parts are both in . It is well known that every function has a non-tangential limit for almost every . We will apply the following theorem concerning harmonic Hardy spaces, see, e.g. [9, p. 15], [8, p. 38]. Theorem B Let , and assume that . Then for almost all tends to a finite limit , as nontangentially, , and, for ,
Let us emphasize that Theorem B does not hold for . In the Remark we give an example of a function which cannot be recovered from its boundary behavior.
We also note that this theorem actually implies an extended version of the maximum (minimum) principle for real harmonic functions in .
Section snippets
Proof of theorem
The following lemma is a modified version of Lemma 2.5 in [7]. Lemma 1 If , then is convex. Proof By Theorem 5.7 in [2], it suffices to show that the function is convex in the direction for every . The function is convex in the direction if and only if is convex in the vertical direction. Let us first assume that . We apply Theorem A with . We have
References (11)
- et al.
Harmonic mappings onto convex domains
Can. J. Math.
(1987) - et al.
Harmonic univalent functions
Ann. Acad. Sci. Fenn. Ser. A I Math.
(1984) Convolutions of planar harmonic convex mappings
Complex Var. Theory Appl.
(2001)- et al.
Convolutions of harmonic convex mappings
Complex Var. Elliptic Equ.
(2012) Harmonic Mappings in the Plane
(2004)
Cited by (13)
ON CONVEX COMBINATIONS of CONVEX HARMONIC MAPPINGS
2017, Bulletin of the Australian Mathematical SocietyDirectional Convexity Of Combinations Of Harmonic Half-Plane And Strip Mappings
2022, Communications of the Korean Mathematical SocietyConvolutions of planar harmonic strip mappings
2021, Complex Variables and Elliptic EquationsConvolutions of a subclass of harmonic univalent mappings
2020, Turkish World Mathematical Society Journal of Applied and Engineering MathematicsConvolutions of Harmonic Right Half-Plane Mappings with Harmonic Strip Mappings
2019, Bulletin of the Malaysian Mathematical Sciences SocietyLinear combinations of harmonic mappings
2018, arXiv