Quasiconformal extension of meromorphic functions with nonzero pole (Q2796729)

The authors use the following notation: \(D = \{z : |z|< 1\}\), \(\partial D = \{z : |z| = 1\}\), \(\overline{D}=\{z:|z|\leq 1\}\), \(D^*=\{z:|z|>1\}\), \(\overline{D}^* = \{z : |z| \geq 1\}\). Let \(f\) be a meromorphic and univalent function in the unit disc \(D\) with a simple pole at \(z=p \in [0, 1)\) of residue 1. Since \(f(z) - 1/(z - p)\) is analytic in \(D\), one has an expression of the form NEWLINE\[NEWLINE f(z) = \frac{1}{z - p}+ \sum_{n=0}^{\infty} a_n z_n,\quad z \in D. \leqno(1) NEWLINE\]NEWLINE We denote the class of such functions by \(\Sigma(p)\). Let \(\Sigma^0(p)\) be the subclass of \(\Sigma(p)\) consisting of those functions \(f\) for which \(a_0 = 0\) in the above expansion. For a given number \(0 \leq k< 1\), \(\Sigma_k(p)\) stands for the class of those functions in \(\Sigma(p)\) which admit a \(k\)-quasiconformal extension to the extended plane \(\hat{\mathbb C}\).NEWLINENEWLINE\textit{O. Lehto} [Ann. Acad. Sci. Fenn., Ser. A I 500, 10 p. (1971; Zbl 0226.30020)] refined the Bieberbach-Gronwall area theorem to the functions in \(\Sigma_k(0)\) in the following form.NEWLINENEWLINETheorem A. Let \(0 \leq k< 1\). Suppose that \(f(z) = 1/z + a_0 + a_1z + a_2 z^2 +\dots\) is a function in \(\Sigma_k(0)\). Then NEWLINE\[NEWLINE \sum_{n=1}^{\infty}n|a_n|^2 \leq k^2.NEWLINE\]NEWLINE Here, equality holds if and only if NEWLINE\[NEWLINE f(z) = 1/z+ a_0 + a_1 z,\quad z \in D, NEWLINE\]NEWLINE with \( |a_1|=k\). Moreover, its \(k\)-quasiconformal extension is given by setting NEWLINE\[NEWLINE f(z) = 1/z+ a_0 +a_1/\overline {z} NEWLINE\]NEWLINE for \(z \in \overline {D^*}\).NEWLINENEWLINEOn the other hand, the area theorem was extended by \textit{P. N. Chichra} [Proc. Camb. Philos. Soc. 66, 317--321 (1969; Zbl 0195.08903)] to functions in \(\Sigma(p)\) as follows.NEWLINENEWLINETheorem B. Let \(f \in \Sigma(p)\) have the expansion in (1). Then NEWLINE\[NEWLINE \sum_ {n=1}^{\infty} n|a_n|^2 \leq1/(1 - p^2)^2 . NEWLINE\]NEWLINE Equality holds for the function NEWLINE\[NEWLINE f_p(z) = 1/(z - p)+ a_0 +z/(1 - p^2) . NEWLINE\]NEWLINE The authors prove the following result.NEWLINENEWLINETheorem. Let \(0 \leq k< 1\) and \( 0 \leq p< 1\). Suppose that \(f\in \Sigma_k(p)\) is expressed in the form of (1). Then NEWLINE\[NEWLINE \sum_{ n=1}^{\infty} n |a_n|^2 \leq k^2/(1 - p^2)^2 . NEWLINE\]NEWLINE Here, equality holds if and only if \(f\) is of the form NEWLINE\[NEWLINE f(z) = 1/(z - p)+ a_0 +a_1 z/(1 - pz) NEWLINE\]NEWLINE for \(z \in D\), where \(a_0\) and \(a_1\) are constants with \(|a_1| = k\). Moreover, a \(k\)-quasiconformal extension of this \(f\) is given by setting NEWLINE\[NEWLINE f(z) = 1/(z - p)+ a_0 +a_1/(\overline{z} - p) NEWLINE\]NEWLINE for \(z \in \overline{D^*}\).