Area distortion under certain classes of quasiconformal mappings

In this paper we study the hyperbolic and Euclidean area distortion of measurable sets under some classes of K-quasiconformal mappings from the upper half-plane and the unit disk onto themselves, respectively.

In  Astala [] proved that if f is a K -quasiconformal mapping from the unit disk D onto itself, normalized by f () = , and if E is any measurable subset of the unit disk, then A e (f (E)) ≤ a(K)A e (E) /K , where A e (·) denotes the Euclidean area and a(K) →  when K →  + .
In  Reséndis and Porter [] obtained some results about area distortion under quasiconformal mappings on the unit disk D onto itself with respect to the hyperbolic measure. They also showed the existence of explodable sets; this kind of sets has bounded hyperbolic area, but under a specific quasiconformal mapping its image has infinite hyperbolic area.
In recent years harmonic quasiconformal mappings have been extensively studied, see [-] and the papers cited therein. The following two recent results are very close to the results presented in this paper.
In  Knežević and Mateljević [] proved the following Schwarz-Pick type distortion theorem.
Theorem  Let f be a K -quasiconformal harmonic mapping from the unit disk D onto itself, then where k is defined by k := K- K+ .
In  Min Chen and Xingdi Chen [] studied the class of (K, K ) quasiconformal maps from H onto itself, and they obtained the following result about area distortion of harmonic mappings.
Theorem  Let f (z) = u(z) + iv(z) be a harmonic mapping from H onto itself and continuous on H ∪ R with f (∞) = ∞. In particular, f has the form f (z) = f (x + iy) = u(x, y) + icy for some c >  (see []). If f is K -quasiconformal and E ⊂ H is any measurable set, then (i) A e (f (E)) ≤ c  KA e (E), In this paper we use the following hyperbolic density definitions: |dz|  -|z|  and |dw| Im w for the unit disk D and the upper half-plane H, respectively, see [] and []. We denote also by A H the hyperbolic area in the unit disk D.

Results and discussion
Our purpose in this article is to continue the study of the hyperbolic area distortion under K -quasiconformal mappings from the upper half-plane H onto itself or from the unit disk D onto itself. Due to the existence of explodable sets (see []), we study some particular classes of quasiconformal mappings. First we apply the result of Knežević and Mateljević [] to estimate the hyperbolic area distortion under harmonic quasiconformal mappings from D onto itself, and we remove the hypothesis of harmonicity in Theorem . Additionally, we generalize even more the class studied by Chen and Chen in []. More precisely, the main result of this paper is the following.
Theorem  Let f be a K -quasiconformal mapping from H onto itself such that f maps a family of horocyclics with a common tangent point onto a family of horocyclics. Then, for each measurable set E ⊂ H, the following inequalities hold: These bounds are asymptotically sharp when K →  + .
Additionally, we obtain some results about radial and angular quasiconformal mappings. Motivated by the generalization mentioned above, we finally describe a set that contains the region of values of the partial derivatives of K -quasiconformal mappings.

Harmonic quasiconformal mappings
In this part we use Theorem  to estimate the hyperbolic area distortion under quasiconformal harmonic mappings from the unit disk onto itself and analyze the hyperbolic and Euclidean area distortion under quasiconformal mappings f (z) = f (x + iy) = u(x, y) + icy, with c > , from H onto itself, without the hypothesis of harmonicity of f (see []), and we sketch the proof of items (i) and (ii) of Theorem  as a corollary of this result and get better bounds than the ones obtained in items (iii) and (iv) in the same theorem. See [] and [] for results in hyperbolic geometry.
Theorem  Let f be a K -quasiconformal harmonic mapping from the unit disk D onto itself. If E ⊂ D is a measurable set, then These bounds are asymptotically sharp when K →  + .
Proof Let w = f (z) = u(z) + iv(z), z ∈ D. By hypothesis and Theorem , the mapping f satisfies Moreover, the Jacobian J f of f satisfies Thus, for each measurable set E ⊂ D, the following equalities hold: The previous sketch gives us also some idea to study quasiconformal mappings of the form f (x + iy) = u(x, y) + iv(y).

Quasiconformal mappings f (x + iy) = u(x, y) + iv(y)
We now generalize the class studied by Cheng and Chen (see []) in two directions. First, we will show that it is possible to avoid the harmonic hypothesis, and second, we will prove that the class of K -quasiconformal mappings given by f (x + iy) = u(x, y) + iv(y) is a family that accepts asymptotically sharp bi-bounds for the area distortion. Let  ≤ K < ∞ and ⊂ C be a domain. Suppose that f : → C is a K -quasiconformal mapping given by () Then f satisfies or equivalently Then f satisfies inequality () if and only if From now on each expression that involves partial derivatives will be true almost everywhere (a.e.) and will denote a domain of the complex plane C.
In this part we focus on K -quasiconformal mappings f from H onto itself given by f (x + iy) = u(x, y) + iv(y). In particular f can be extended homeomorphically to H, u(x, y) is ACL and v(y) is absolutely continuous. We know that f satisfies Despite the fact that v depends only on the variable y, we write v y instead of v to emphasize the dependence on y. The next result gives the principal characteristics of the mapping f , and most of them are consequences of its quasiconformal properties (the rest are easy to prove).  We study inequality () in more detail. To get this, we complete in () the square in v y , so we obtain This inequality defines a circle a.e., thus u x and v y satisfy in particular In fact, the circle is a subset of the square described by the previous inequalities. Observe that where K is the maximal dilatation of f . Let C = √ α  -. Then  ≤ C. With this notation the last inequalities can be written as follows: and -Cv y ≤ u y ≤ Cv y a.e. () If we choose any fixed y ∈ (, ∞) such that u(x, y) is absolutely continuous with respect to x, then we obtain for each x ∈ R and almost every y ∈ (, ∞). Using the left-hand side of the last inequality, we get and since u(x, y) is continuous, we obtain For this reason, lim sup y→ + v y (y) exists and consequently lim inf y→ + v y (y) exists too. We On the other hand, we choose any fixed x ∈ [, ∞) such that u(x, y) is absolutely continuous with respect to y, and we integrate () on the interval [, y]. So and, since v(y) is absolutely continuous, we obtain for y ∈ [, ∞) and almost every x ∈ [, ∞). By an argument of continuity of the mapping f and density, we have . Setting x =  in the previous inequality, we get Thus, combining () and (), we have for each x ∈ R and almost every y ∈ (, ∞). In the same way we use () and () to obtain We combine the left-and right-hand sides of the previous inequalities to get for each x ∈ R and almost every y ∈ (, ∞). Since the left-and right-hand sides of inequalities () and () represent linear equations in the variable x, we compare their slopes and the fact that for each x ∈ R and almost every y ∈ (, ∞). Hence for each x ∈ R and almost every y ∈ (, ∞). We recall that v(y) is absolutely continuous, and we integrate the above inequalities on the interval [, y] and v has right Dini's derivatives at . If x < , we obtain the same relation as () and have the next result.
Theorem  Let f be a K -quasiconformal mapping from H onto itself given by f (x + iy) = u(x, y) + iv(y). Then v * y () = lim sup y→ + v y (y) and vy () = lim inf y→ + v y (y) are finite, and the partial derivatives of f satisfy the following inequalities: In particular by Proposition  the partial derivatives of f belong to some truncate solid cone.
We can combine the previous result with items (iii) and (iv) of Theorem  to obtain the following result.

Corollary  Let f be a harmonic K -quasiconformal mapping from
where a = . . . . and b = . . . . are the solutions of the equations K  = Proof The left-hand side of the first inequality is immediate from item  of Theorem . We now consider the right-hand side estimations of u Then We consider the equation and obtain the value of a by solving K  -K  -K  +  =  that has only two real solutions K =  and K = . . . . = a. We now consider the estimations of |u y | in item  of Theorem  and item (iv) of Theorem . Thus The value of b is given by the real solution of that can be reduced to the equation and the real solution of this equation is K = . . . . = b. This value is obtained through the equation discarding the value K = . . . . that obviously is not a solution of the original equation.
The last corollary shows that these estimations are asymptotically sharp when K →  + improving the result obtained in [].
We estimate the last expression using Theorem  to obtain . Let z  , z  ∈ H and l be the hyperbolic segment that joints z  with z  . Then and these inequalities are asymptotically sharp when K →  + .
Proof Let E ⊂ H be a measurable set. The Jacobian of f is J f = u x v y . By () and item  of Theorem , The Euclidean area of f (E) is and in consequence On the other hand, for the hyperbolic area only, we see that from () it follows Now, we can prove Theorem .
Proof of Theorem  If f : H → H is a K -quasiconformal mapping that leaves invariant the family of horocyclics with common tangent point at ∞, then f (x + iy) = u(x, y) + iv(y) and is exactly Theorem .
We now suppose that f : H → H is a K -quasiconformal mapping such that f maps horocyclics with tangential point at x  ∈ R onto horocyclics with tangential point at x  ∈ R.
For i = , , let H i be a horocyclic with tangential point at x i and H i ∞ ⊂ H be a horocyclic with tangential point at infinity. Then there exist Möbius transformations T i such that , thenf is K -quasiconformal and can be written in the formf (x + iy) = u(x, y) + iv(y). By Theorem  we have that if E ⊂ H is a measurable set, thenf satisfies Since the hyperbolic area in H is invariant under the Möbius transformations T i , the result follows immediately. The other case is similar.
Corollary  Let f be a K -quasiconformal mapping from H onto itself given by f (x + iy) = u(x, y) + iv(y). Then, for each measurable set E ⊂ H, These inequalities are asymptotically sharp when K →  + .
Proof If v is differentiable at , then from () we get vy () If v is continuously differentiable in a neighborhood of , then vy () = v y () = v + y (). We prove the corollary applying item  of Theorem .
The following examples of quasiconformal mappings are not harmonic, thus we are generalizing the results obtained in [].

Example 
. Let f : H → H given by Thus, for each measurable set E ⊂ H, we have and . Let f : H → H given by f (x + iy) = x + sin(x + y) + iy. Then f is a It is enough to observe that the dilatation of f is given by = cos  (x + y) + ( + cos(x + y))  + cos  (x + y) + ( + cos(x + y))  cos  (x + y) + ( + cos(x + y))  -cos  (x + y) + ( + cos(x + y))  and the function D f (x, ax) depends only on a, more precisely The maximum of D f (x, ax) is attained at the critical points a = lπ , l = , ±, ±, . . . and the maximal dilatation of f is Thus, for each measurable set E ⊂ H, we have

Angular and radial quasiconformal mappings
In this part we obtain the results of area distortion for radial and angular mappings. In the case of angular mappings, we use the hyperbolic model of the unit disk D.
Proposition  Let f : → C be an ACL mapping. If f (re iθ ) = u(re iθ ) + iv(re iθ ), then for a.e. in and the Jacobian of f is A mapping f : H → H is said to be radial at x ∈ R if f leaves invariant all Euclidean rays in H that meet at x.
Let f : H → H be a radial mapping at x. Since hyperbolic area is invariant under horizontal translations, we can assume that the point x ∈ R, where the Euclidean rays meet, is x = . If f is a radial mapping, then f can be written in polar coordinates (r, θ ) as Lemma  Let f be an ACL mapping from H onto itself. Suppose that f is a radial mapping at . Then its Jacobian is J f = ρρ r r a.e. If f preserves orientation, then ρ r >  a.e.
By an argument of continuity of f and density, we finally obtain In a similar way, if we suppose that  < R < , then Theorem  Let f be a K -quasiconformal mapping from H onto itself that leaves invariant each ray in H that meets a real base point. If E ⊂ H is a measurable set, then These inequalities are asymptotically sharp when K →  + .
or equivalently, in rectangular coordinates, that is, Example  Let f : H → H be the √ + √ - -quasiconformal mapping which is radial at  given by Then, for all measurable set E ⊂ H, the mapping f satisfies In the following case we consider the hyperbolic model in the unit disk D. A mapping f : D → D is said to be angular at  ∈ D if f leaves invariant each circle in D with center at .
An angular mapping f at  can be written as f (z) = f (re iθ ) = re iϕ(r,θ) , where ϕ : Lemma  Let f be an ACL mapping from D onto itself. Suppose that f is angular at . Then its Jacobian is J f = ϕ θ . If f preserves orientation, then ϕ θ >  a.e.
Proposition  Let f be a K -quasiconformal mapping from D onto itself which is angular at . Then Proof If f (z) = f (re iθ ) = re iϕ(r,θ) , from () and () we get Since we get the result.
Theorem  Let f be a K -quasiconformal mapping from D onto itself which is angular at . If E ⊂ H is a measurable set, then These inequalities are asymptotically sharp when K →  + .
Proof Let E ⊂ D be a measurable set and E denote the set E in polar coordinates. If f : D → D is given as before by f (z) = re iϕ(r,θ) , then or equivalently, in rectangular coordinates, and this concludes the proof.
Then f is a π  + π +  + e π -quasiconformal mapping. Thus, for each measurable set E of the unit disk D, the following inequalities hold: Numerical evidence says that the maximal dilatation of f can be e π .
The following example shows that the result of Theorem  can not be generalized to radial mappings at .
Example  Let K ≥ . Let f , g : D → D be the K -quasiconformal mappings given by f (re iθ ) = r  K e iθ and g(re iθ ) = r K e iθ . For each r ∈ (, ), define E r = {z ∈ D : |z| < r}. Then Then if K > , there is not C >  such that for all r ∈ (, ).

The Beurling-Ahlfors extension
Using the Beurling-Ahlfors (BA) extension, we give explicit examples of quasiconformal mappings of the form f (x, y) = u(x, y) + iv(x, y) and their associated bi-bounds for the hyperbolic area distortion. More precisely, let h : R → R be an increasing homeomorphism and define its Beurling- for all x ∈ R and t > . It is well known that its BA-extension is a K = K(M) ≥  quasiconformal mapping, even more; Ahlfors proved in [] that this extension is a quasi-isometry, that is, there exists Then h is a -quasisymmetric function and its BA-extension is the -quasiconformal and harmonic mapping defined by f (x + iy) = x + iy  . Moreover, for each measurable set E ⊂ H, Example  Let g(x) = x  . Then g is a  +  √ -quasisymmetric function, and its BAextension is the . . . . quasiconformal mapping f (x + iy) = x  + xy  + i  (x  y + y  ). In particular f (x, y) does not have bounded derivatives. Moreover, Setting y = cx in the right-hand side, we get for x =  r(c) = ( -c  + c  ) ( + c  )  and it is easy to see that   ≤ r(c) <  for each c ∈ R.
Thus, for each measurable set E ⊂ H, we have We have that A e (E) =  and, since J f (x + iy) =   (x  -x  y  + y  ), it holds A e (f (E)) = ∞. Thus f explodes Euclidean area.
Example  Let g(x + iy) = x + sin(x + y) + iy. Then g is a + √   -quasiconformal mapping from H onto itself. Thus the function h(x) = x + sin x is . . . . -quasisymmetric and its BA-extension is . . . . -quasiconformal, given by In a forthcoming paper we study more deeply quasi-isometric properties of the BA extension.

A set that contains the region of values of the partial derivatives of K-quasiconformal mappings
In this part we study some particular forms of the mapping f in (). First, suppose that f is a K -quasiconformal mapping given by f (x + iy) = u(x) + iv(y). Then, by (), its partial derivatives satisfy the inequality Since α ≥ , the discriminant of u  x + v  y -αu x v y is non-negative and () defines the interior of an angular region with the identification u x ∼ x -axis and v y ∼ y -axis. Thus we have proved.
Theorem  Let  ≤ K < ∞. If f : → C is a K -quasiconformal mapping given by f (x + iy) = u(x) + iv(y), then its partial derivatives belong to one of the angular regions defined by ().
Proof The proof follows from the fact that the Jacobian of f is always positive.
If f is a K -quasiconformal mapping given by f (x + iy) = u(x, y) + iv(y), then () reduces to Inequality () suggests studying the quadratic form Q(x, y, w) = x  + y  + w  -αxw, whose associated symmetric matrix is

Proposition  There exists an invertible matrix P such that P
Proof The eigenvalues of N are λ  = α, λ  =  and λ  =  + α with eigenvectors (, , ), (, , ) and (, , -), respectively. After normalization we obtain the basis B := A simple calculus ends the proof.
In Proposition  Let  < K < ∞. If f : → C is a K -quasiconformal mapping given by f (x + iy) = u(x, y) + iv(y), then its partial derivatives u x , u y and v y belong to one branch of the elliptic cone ().
Proof As we saw f is K -quasiconformal if and only if u x , u y and v y satisfy Q(u x , u y , v y ) ≤  that describes the solid cone (). As f preserves orientation, then J f = u x v y >  a.e. Since v y >  a.e., then necessarily u x >  a.e., and the result follows.
We do not study the case f (x + iy) = u(x, y) + iv(x) because this kind of mapping is not a homeomorphism from H onto itself and in consequence is not quasiconformal.
We consider now the general case, that is, a quasiconformal mapping f : ⊂ C → C given by f (x + iy) = u(x, y) + iv(x, y), then by () we have that In this case we study the quadratic form Q(x, y, z, w) = x  + y  + z  + w  -αxw + αyz with the associated symmetric matrix Proposition  There exists an invertible matrix P such that P - NP = D, where Proof The characteristic polynomial of the matrix N is (λ) α  (λ) α  (λ)  + α  with eigenvalues λ  =  + α, λ  = α, and both with multiplicity two. The eigenvectors of λ  are (, , , -) and (, , , ) and for λ  are (, , , ) and (, , -, ). After normalization we obtain the matrix Thus P - NP = D.

Corollary  The quadratic form
Q( x, y, z, w) = (α) x  + (α) y  + ( + α) z  + ( + α) w  represents the quadratic form Q(x, y, z, w) = x  + y  + z  + w  -αxw + αyz As it is showed in Theorems , , , ,  and Corollaries  and , we have obtained left and right asymptotic bounds for the hyperbolic or Euclidean area distortion. In some previous results only right bounds were known. Moreover, the examples showed that the different classes of mappings defined in the paper are not empty.
We do not know whether the branch of the elliptic cone (), mentioned in Proposition , coincides or not with the region of variation of the partial derivatives of quasiconformal mappings f (z) = u(x, y) + iv(y).