1 Introduction and Preliminaries

In response to the classical Grötzsch problem raised in 1928, Ahlfors introduced the notion so-called “quasiconformal mappings” in 1935. Quasiconformal mappings are nothing but generalizations of conformal mappings. There are several equivalent definitions of quasiconformal mappings in the literature (see instance [1, 8]). In this paper, we adopt the following definition of Ahlfors. Let \(K\ge 1\). A homeomorphism f is called K-quasiconformal if f has locally \(L^2\)-derivative and it satisfies the Beltrami differential equation \(f_{\overline{z}}(z) = \mu (z)f_z(z)\) a.e., where \(\mu \) satisfies

$$\begin{aligned} |\mu (z)| \le \frac{K-1}{K+1}=k<1; \end{aligned}$$
(1.1)

\(f_{\overline{z}}=\partial f /\partial \overline{z}\) and \(f_z=\partial f /\partial z\). The function \(\mu \) is called the complex dilatation of f. Note that f is conformal if and only if \(\mu \) vanishes identically. Therefore, 1-quasiconformal mappings are nothing but conformal. For basic properties of quasiconformal mappings, we refer to [8].

By \(\Sigma \), we denote the class of functions of the form:

$$\begin{aligned} g(z)=z+b_0+\frac{b_1}{z}+\cdots \end{aligned}$$
(1.2)

that are analytic and univalent in the domain \(\Omega :=\{z:|z|>1\}\), except for simple pole at infinity with residue 1. The class \({\Sigma }'\) denotes the collection of functions g in \(\Sigma \), such that \(g(z)\ne 0\) in \(\Omega \). Using a simple geometric argument, Gronwall [6] in 1914 proved the classical area theorem, which says that the coefficients of \(g \in \Sigma \) satisfy the sharp inequality \(\sum n |b_n|^2\le 1\). Furthermore, Lehto [7] generalized the area theorem by assuming the additional hypothesis that g admits a quasiconformal extension to the closed unit disk, where the resultant inequality is sharp. For updated research work related to the area theorem, readers can refer to [2, 3]. Closely related to the class \(\Sigma \) is the class \({\mathcal {S}}\) of all analytic univalent functions f defined in the unit disk \({\mathbb {D}}:=\{z:|z|<1\}\) with the normalization \(f(0)=0\) and \(f'(0)=1\). Note that functions in \({\mathcal {S}}\) have power series representation of the forms:

$$\begin{aligned} f(z)=z+a_2z^2+\cdots . \end{aligned}$$
(1.3)

It is easy to verify that each \(f \in {\mathcal {S}}\) is associated with a function \(g\in \Sigma '\) through the relation \(g(z)=\{f(1/z)\}^{-1}\). Therefore, there exists a one-to-one correspondence between \({\mathcal {S}}\) and \(\Sigma '\) (see [4, p 28]).

For an analytic function f in \({\mathbb {D}}_r:=\{z:|z|<r,\,0<r\le 1\}\), we set

$$\begin{aligned} \Delta (r,f)=\iint _{{\mathbb {D}}_r} |f'(z)|^2 \mathrm{d}x \mathrm{d}y, \quad z=x+i y, \end{aligned}$$
(1.4)

which is called the Dirichlet integral of f. Geometrically, this describes the area of the image of \({\mathbb {D}}_r\) under f. One of the classical problems in univalent function theory is to obtain the class of functions f having finite Dirichlet integral \(\Delta (1,f)\); we call such functions f as Dirichlet finite. In the recent years, such problems for various subclasses of \({\mathcal {S}}\) have been studied by Ponnusamy and his co-authors; see, for instance, [9,10,11,12,13]. The motivation to study these problems comes from a conjecture of Yamashita [14] which is settled in [9]. In this paper, we extend the problem of Yamashita to the functions in the family \({\mathcal {S}}\) having quasiconformal extension to the entire complex plane.

Let k be defined as in (1.1). We denote \(\Sigma (k)\) by the class of all functions \(g \in \Sigma \) that admit K-quasiconformal extension to the unit disk \({\mathbb {D}},\) and \(\Sigma _0(k)\) is obtained from \(\Sigma (k)\) by assuming \(g(0)=0.\) Similarly, let us denote \({\mathcal {S}}(k)\) by the class of all functions \(f \in {\mathcal {S}}\) that admit K-quasiconformal extension to the plane. Clearly, \(f \in {\mathcal {S}}(k)\) if and only if \(1/f(1/\zeta ) \in \Sigma _0(k).\)

Rest of the paper is organized as follows. Some well-known key results are collected in Sect. 2 followed by the proof of our main theorem. We observe that the modified Koebe function studied in [7] does not play extremal role in our problem. However, we construct a new function which also extends the Koebe function \(z/(1-z)^2\) to the K-quasiconformal setting and show that it plays the extremal role in our problem. Section 3 is devoted to the comparison of the areas obtained in Sect. 2 for our extremal function with the modified Koebe function.

2 Main Result

Suppose that f is an analytic function in the disk \({{\mathbb {D}}}\) with the Taylor series expansion:

$$\begin{aligned} f(z)=\sum _{n=0}^\infty a_nz^n \end{aligned}$$
(2.1)

and \(f'(z)=\sum _{n=0}^\infty n a_nz^{n-1}.\) Then, using Parseval–Gutzmer formula, the area \(\Delta (r,f)\), of \(f(\overline{{\mathbb {D}}}_r)\), as stated in (1.4) can be re-formulated as follows (see [5]):

$$\begin{aligned} \Delta (r,f)=\iint _{{\mathbb {D}}_r} |f'(z)|^2 \mathrm{d}x \mathrm{d}y=\pi \sum _{n=1}^\infty n|a_n|^2r^{2n},\quad z=x+iy. \end{aligned}$$
(2.2)

We concentrate particularly on this form of the area formula in this paper. Computing this area is called the area problem for the functions of the typef. However, area of \(f({\mathbb {D}})\) may not be bounded for all \(f \in {\mathcal {S}}\). We remark that if \(f\in {\mathcal {S}}\), then z / f is non-vanishing, and hence, \(f\in {\mathcal {S}}\) may be expressed as follows:

$$\begin{aligned} f(z)=\frac{z}{F_f(z)}, \quad \text{ where } F_f(z)=1+\sum _{n=1}^\infty c_nz^n, \quad z\in {\mathbb {D}}. \end{aligned}$$

Yamashita in [14] considered the area problem for functions of type \(F_f\) for \(f \in {\mathcal {S}},\) and proved that the area of \(F_f({\mathbb {D}}_r)\) is bounded. Indeed, he proved.

Theorem A

[14, Theorem 1] We have

$$\begin{aligned} \max _{f\in {\mathcal {S}}} \Delta (r,F_f)=2\pi r^2(r^2+2), \end{aligned}$$

for \(0<r\le 1.\) The maximum is attained only for a suitable rotation of the Koebe function.

To consider the Yamashita problem for functions in \({\mathcal {S}}\) having quasiconformal extension to the entire complex plane, the following theorem of Lehto [7] is useful.

Theorem B

Let \(g \in \Sigma (k)\) be of the form (1.2). Then

$$\begin{aligned} \sum _{n=1}^\infty n |b_n|^2 \le k^2. \end{aligned}$$
(2.3)

The equality holds for the function:

$$\begin{aligned} g(z)=\frac{1}{z}+a_0+a_1z, \quad z\in {\mathbb {D}}, \end{aligned}$$

with \(|a_1|=k.\) Moreover, its k-quasiconformal extension is given by setting:

$$\begin{aligned} g(z)=\frac{1}{z}+a_0+\frac{a_1}{\overline{z}}, \quad z\in \overline{\Omega }. \end{aligned}$$

We also need an immediate consequence of Theorem B, proved by Lehto in the same paper, which gives the sharp bound for second coefficient of functions in \({\mathcal {S}}\) having quasiconformal extension to the plane. The consequence is stated as follows:

Theorem C

[7, Corollary 3] For a function \(f \in {\mathcal {S}}(k)\) of the form (1.3) with \(f(\infty )=\infty \), we have \(|a_2|\le 2k.\)

Using Theorem B and Theorem C, we now state and prove our main result.

Theorem 2.1

For \(0<r\le 1,\) we have

$$\begin{aligned} \max _{f\in {\mathcal {S}}(k)} \Delta (r,F_f)=2\pi r^2 k^2 (2+r^2). \end{aligned}$$

The maximum is attained only for a suitable rotation of the function:

$$\begin{aligned} f(z)= \left\{ \begin{array}{ll} \displaystyle \frac{z}{1-2kz+kz^2}, &{} \quad { \text{ for } }\; |z|< 1, \\ \displaystyle \frac{z\overline{z}}{\overline{z}-2 k z\overline{z}+kz} , &{} \quad { \text{ for } }\; |z|\ge 1. \end{array} \right. \end{aligned}$$
(2.4)

Proof

Let \(f\in {\mathcal {S}}(k)\) be of the form (1.3). Then

$$\begin{aligned} \frac{1}{f(\frac{1}{z})}=z-a_2+(a_2^2-a_3)\frac{1}{z}+\cdots = z+b_1+\frac{b_2}{z}+\cdots \,\, (\text{ say }). \end{aligned}$$

Substituting 1 / z by z and multiplying z, we obtain

$$\begin{aligned} F_f(z)=\frac{z}{f(z)}=1-a_2 z+(a_2^2-a_3)z^2+\cdots = 1+b_1 z+b_2 z^2+\cdots \end{aligned}$$

It is clear that \(b_1=-a_2\). Now, we compute

$$\begin{aligned} \frac{1}{\pi }\Delta (r,F_f)= & {} \sum _{n=1}^\infty n|b_n|^2r^{2n}\\= & {} |b_1|^2 r^2+\sum _{n=2}^\infty n|b_n|^2r^{2n}\\= & {} |-a_2|^2 r^2+2r^4 \sum _{n=1}^\infty \frac{n+1}{2}|b_{n+1}|^2r^{2n-2}. \end{aligned}$$

Using the estimate for \(a_2\) from Theorem C, we obtain

$$\begin{aligned} \frac{1}{\pi }\Delta (r,F_f)\le 4r^2k^2+2r^4\sum _{n=1}^\infty n|b_{n+1}|^2. \end{aligned}$$

Then, by Theorem B, we have

$$\begin{aligned} \frac{1}{\pi }\Delta (r,F_f)\le 4r^2k^2+2r^4k^2=2r^2k^2(r^2+2). \end{aligned}$$

Now, it remains to consider the sharpness part. For \(|z|< 1\), consider the function \(f(z)=z/(1-2kz+kz^2)\). Therefore, \(f_{\overline{z}}=0\). That is, f is conformal in \({{\mathbb {D}}}\). Since \(F_f(z)=1-2kz+kz^2\), by (2.2), we obtain

$$\begin{aligned} \frac{1}{\pi }\Delta (r,F_f)=\sum _{n=1}^\infty n|b_n|^2r^{2n}=4r^2k^2+2r^4k^2=2r^2k^2(r^2+2). \end{aligned}$$

For \(|z|\ge 1\), let

$$\begin{aligned} f(z)=\frac{z\overline{z}}{\overline{z}-2 k z\overline{z}+kz}. \end{aligned}$$

An easy calculation shows that

$$\begin{aligned} f_{\overline{z}}=\frac{z(\overline{z}-2kz\overline{z}+kz)-z\overline{z}(1-2kz)}{(\overline{z} -2kz\overline{z}+kz)^2}=\frac{kz^2}{(\overline{z}-2kz\overline{z}+kz)^2} \end{aligned}$$

and

$$\begin{aligned} f_z=\frac{\overline{z}(\overline{z}-2kz\overline{z}+kz)-z\overline{z}(-2k\overline{z}+k)}{(\overline{z}-2kz\overline{z}+kz)^2}=\frac{\overline{z}^2}{(\overline{z}-2kz\overline{z}+kz)^2}. \end{aligned}$$

Thus, \(|f_{\overline{z}}/f_{z}|=k\).

Both the functions defined in (2.4) agree on the boundary \(\partial {{\mathbb {D}}}\) of \({{\mathbb {D}}}\). The proof is complete. \(\square \)

Remark 2.2

Observe that Theorem 2.1 is a natural extension of Theorem A. In fact, for \(k=1,\) Theorem 2.1 is equivalent to Theorem A.

Remark 2.3

It is easy to check that for \(f \in {\mathcal {S}}(k),\, \Delta (1,F_f)\le 6\pi k^2\), and hence, \(F_f\) is Dirichlet finite.

Table 1 Comparison of areas of \(F_f({\mathbb {D}})\) and \(F_g({\mathbb {D}})\)
Full size table

3 Comparison of Areas

Recall the modified Koebe function from [7] which is defined by the following:

$$\begin{aligned} g(z)= \left\{ \begin{array}{ll} \frac{z}{(1+ke^{i\phi }z)^2}, &{} \quad { \text{ for } }\, |z|< 1, \\ \frac{z\overline{z}}{(\sqrt{\overline{z}}+ke^{i\phi }\sqrt{z})^2}, &{}\quad { \text{ for } }\, |z|\ge 1. \end{array} \right. \end{aligned}$$
(3.1)

A simple computation yields

$$\begin{aligned} \Delta (r,F_g)=2r^2k^2(k^2r^2+2)\pi , \end{aligned}$$

which geometrically describes the area of \(F_g({\mathbb {D}})\). Note that

$$\begin{aligned} 2r^2k^2(k^2r^2+2)\pi =\Delta (r,F_g)<\Delta (r,F_f)=2r^2k^2(r^2+2). \end{aligned}$$

To see the graphical and numerical comparisons of the Dirichlet finites \(\Delta (1,F_g)\) and \(\Delta (1,F_f)\), we end this section with the following observations (Table 1; Figs. 1, 2, 3, 4). First, we show the graphs of \(F_f\) and \(F_g\), where f and g are defined by (2.4) and (3.1) respectively, for different values of k. Here, the terminology the graph of\(F_f\) means the image domain\(F_f({\mathbb {D}})\) and, similarly, for the graph of \(F_g\). Observe that as \(k \rightarrow 1\), the graphs of \(F_g\) are approaching to those of \(F_f\).

Fig. 1
figure 1

Graphs of \(F_f\) and \(F_g\) for \(k=0.2\)

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Fig. 2
figure 2

Graphs of \(F_f\) and \(F_g\) for \(k=0.5\)

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Fig. 3
figure 3

Graphs of \(F_f\) and \(F_g\) for \(k=0.7\)

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Fig. 4
figure 4

Graphs of \(F_f\) and \(F_g\) for \(k=0.9\)

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Second, for these choices of k, Table 1 compares the area \(\Delta (1,F_g)\), of the image of \({\mathbb {D}}\) under \(F_g\), and the area \(\Delta (1,F_f)\), of the image of \({\mathbb {D}}\) under \(F_f\).