Radii of covering disks for locally univalent harmonic mappings
For a univalent smooth mapping of the unit disk of complex plane onto the manifold , let be the radius of the largest univalent disk on the manifold centered at (). The main aim of the present article is to investigate how the radius varies when the analytic function is replaced by a sense-preserving harmonic function . The main result includes sharp upper and lower bounds for the quotient , especially, for a family of locally univalent -quasiconformal harmonic mappings on . In addition, estimate on the radius of the disk of convexity of functions belonging to certain linear invariant families of locally univalent -quasiconformal harmonic mappings of order is obtained.
Key words and phrases:Locally univalent harmonic mappings, linear and affine invariant families, convex and close-to-convex functions, and covering theorems.
2000 Mathematics Subject Classification:Primary: 30C62, 31A05; Secondary: 30C45,30C75
1. Introduction and Main results
Let be the unit disk, and be a smooth univalent mapping of the unit disk onto two-dimensional manifold . For a point , we write as the radius of the largest univalent disk centered at on the manifold . Here a univalent disk on centered at means that maps an open subset of containing the point univalently onto this disk.
The question about lower estimation of for univalent analytic functions first was considered in papers of Koebe  and Bieberbach  in connection with the well known problem of covering disk in the class . Here denotes the classical family of all normalized univalent (analytic) functions in investigated by a number of researchers (see [12, 14, 21]). In the class of analytic functions in with , the determination of the exact value of the greatest lower bound of all is one of the most important problems in geometric function theory of one complex variable. For historical discussion of the attempts of various mathematicians to estimate the lower bound for , we refer to  and also [4, 6, 7] for recent developments.
If denotes the family of functions analytic and locally univalent in , then the classical Schwarz lemma for analytic functions gives the following well-known sharp upper estimate for :
Often the right hand side quantity, namely, is referred to as the conformal radius of the domain at . Sharp and nontrivial lower estimate for was obtained by Pommerenke  in a detailed analysis of what is called linear invariant families of locally univalent analytic functions in . Throughout we denote by , the set of all conformal automorphisms (Möbius self-mappings) , where and , of the unit disk .
(cf. ) A non-empty collection of functions from is called a linear invariant family (LIF) if for each , normalized such that , the functions defined by
belong to for each .
The order of the family is defined to be . The universal LIF, denoted by , is defined to be the collection of all linear invariant families with order less than or equal to (see ). An interesting fact about the order of a LIF family is that many properties of it depend only on the order of the family. It is well-known  that if and only if . The family is precisely the family of all normalized convex univalent (analytic) functions whereas .
Note that is the largest LIF of functions with the restriction of growth (see ):
In , Pommerenke has proved that for each the following sharp lower estimate of holds:
In the present paper we obtain estimate of the functional when instead of analytic functions we consider harmonic locally univalent mappings
i.e. when is added to the functions . In the above decomposition of , the functions and are called the analytic and co-analytic parts of , respectively. We say that a harmonic functions is sense-preserving if the Jacobian of is positive. Lewy’s theorem  (see also for example [13, Chapter 2, p. 20] and ) implies that every harmonic function on is locally one-to-one and sense-preserving on if and only if in . Note that in if and only if and there exists an analytic function in such that
where . Here is referred to as the (complex) dilatation of the harmonic mapping . When it is convenient, we simply use the notation instead of .
There are different generalizations of the notion of the linear invariant family to the case of harmonic mappings. For example, the question about a lower estimate of the radius of the univalent disk centered at the origin was examined by Sheil-Small  in the linear and affine invariant families of univalent harmonic functions . There are a number of articles in the literature proving such inequalities or studying the related mappings in various settings. For example, see [4, 5, 6, 7, 8, 9, 10, 27, 29], and also the work from  in which one can obtain a lower bound on the radius for quasi-regular mappings. The concept of linear and affine invariance was also discussed by Schaubroeck  for the case of locally univalent harmonic mappings.
The family of locally univalent sense-preserving harmonic functions in the disk of the form (1.1) is called a linear invariant family (LIF) if for each the following conditions are fulfilled: and
for each . A family is called linear and affine invariant (ALIF) if it is LIF and in addition each satisfies the condition that
The number is known as the order of the ALIF .
The order of LIF without the assumption of affine invariance property is defined in the same way: .
Throughout the discussion, we suppose that the orders of these families, namely, and , are finite. The universal linear and affine invariant family, denoted by , is the largest ALIF of order . Thus, the subfamily of ALIF consists of all functions such that If is univalent in , then according to the result of Sheil-Small  one has the following sharp lower estimate:
For and , denote by the set of all locally univalent -quasiconformal harmonic mappings in of the form (1.1) with the normalization such that
Note that the classes , which expand with the increasing values of and , cover all sense-preserving locally quasiconformal harmonic mappings with the indicated normalization.
We shall restrict ourselves to the case of finite . In [27, 28], it was also shown that the family possess the property of linear invariance in the following sense: for each and for every , the transformation
where denotes the directional derivative of the complex-valued function in the direction of the unit vector .
In , Starkov proved that for each and ,
which is equivalent to
and the lower estimate is sharp in contrast to the upper one.
One of the main aims of this article is to establish sharp estimations of the ratio for -quasiconformal harmonic mappings . In particular, sharp upper estimate in (1.5) is obtained. The ratio demonstrates how the radius of the largest univalent disk with the center at on the manifold varies if we add, to the analytic function , the function .
We now state our first result.
Let for some , and be the complex dilatation of the mapping . Then for ,
where . Here the functions and are defined as follows:
where K denotes the (Legendre) complete elliptic integral of the first kind given by
and . The argument is sometimes called the modulus of the elliptic integral .
Suppose that , and , where is a complex constant. Then the following relations hold:
Moreover, after appropriate normalization, every -quasiconformal harmonic mapping in belongs to the family for some . Therefore an equivalent formulation of Theorem 1 may now be stated.
These inequalities and (1.6) give the following.
Let and . Then we have
The sharpness of the last double-sided inequalities at the point follows from the proof of Theorem 1.
We now state the remaining results of the article.
Let be a locally quasiconformal harmonic mapping belonging to the family with , and . Then
The estimation is sharp for example in the universal ALIF .
Recall that a locally univalent function is said to be convex in the disk if maps univalently onto a convex domain. The radius of convexity of the family of functions defined on the disk is the largest number such that every function is convex in the disk .
If , then for every , the function is convex in the disk , where
In particular, the radius of convexity of the family is no less than .
2. Proofs of the Main results
2.1. Proof of Theorem 1
The proof of the theorem is divided into three parts.
Part 1: Let satisfy the assumptions of Theorem 1. In compliance with the definition of the value , there exists a boundary point of the manifold such that . Consider the smooth curve , namely, the preimage of the semi-open segment with the starting point in the disk . Then
where the minimum is taken over all smooth paths , such that and
Similarly we define the value
where the simple smooth curve is emerging from the origin, the preimage of the semi-open segment under the mapping . Consider the following parametrization of the curve : . Then and
At first we consider the case . Since for , we have
If for , then we have the inequality
which proves the upper estimate in the inequality (1.6) for .
Let us now assume that for some . Then, from a generalized version of the classical Schwarz lemma (see for example [14, Chapter VIII, §1]), it follows that
Consequently, by (2.1), one has
Also, the function maps biholomorphically onto some subdomain of the disk . Applying the classical Schwarz lemma, we obtain the inequality and hence, holds. Using the last estimate and the inequality (2.3), one can obtain, after evaluating the integral, the inequality
which is positive, since . Hence
We now begin to prove that the upper estimate in (1.6) is true for all As mentioned above, the family is linear invariant in the sense of [27, 28] (see (1.4) above). Hence, for each fixed , the function defined by
belongs to the family , where and are analytic in such that Therefore, in view of (2.4) for , we have
where if , and when . Note that
and we complete the proof of the upper estimate in (1.6).
Part 2: We now deal with the sharpness of the upper estimate in (1.6). Consider the case . For every and every , we shall indicate functions from the families such that , where Since the families are enlarging with increasing values of , the sharpness of the upper estimate in (1.6) will be shown for every and each .
Consider the sequence of functions from defined by
Then we have (see ) and observe that maps the unit disk univalently onto the Riemann surface whose boundary described by
consists of two rays. Then the univalent image of the disk under the mapping
() represents the manifold with the boundary
which consists of two rays parallel to the coordinate axes and arising from the point . Note that the function maps the semi-open segment bijectively onto and thus, we conclude that
where . The sharpness of the upper estimate in (1.6) is proved for and .
Next we let , , and consider a conformal automorphism of the unit disk . Then the inverse mapping is given by . From the condition (1.4) of the linear invariance property of the family , it follows that the function defined by
belongs to , where and have the same meaning as above. Taking into account of the normalization condition for functions in the family , we deduce that
On the other hand, a direct computation gives
which completes the proof of the upper estimation in Theorem 1 for .
If then for , we consider the sequence of functions
We see that for each . Therefore,
The sharpness of the upper estimation in (1.6) for , can be proved analogously. So, we omit the details.
The images of polar grid in the unit disk under mappings and are indicated in Figures 1(a)-(b) which
illustrate the sharpness assertions proved in the above estimations.
Part 3: Finally, we deal with lower estimation of . If , then the lower estimation in (1.6) is trivial because . So, we may assume that . As in Part 1, we define the boundary points and of the manifolds and , respectively, and smooth curves and in the same manner as in Part 1. Consider the parametrization of the curve :
Then and thus,
In view of the inequality (2.2), we find that
It is possible to obtain an estimate for , with the help of the analog of the Schwarz lemma for -quasiconformal automorphisms of the disk. Let be a -quasiconformal automorphism of , and . It is known (see for example [1, Chapter 10, equality (10.1)]) that the sharp estimation
holds, where and are as in the statement. The function defined on the unit disk satisfies the conditions and . Let be the univalent conformal mapping of the domain onto the unit disk and Then the composition