Univalence and Quasiconformal Extension of an Integral Operator

Theorem 2.1. [25, Pommerenke] Let $\mathcal{L}(z,t) = a_1(t)z + a_2(t)z^2 + ...$ be analytic in $\mathcal{U}_r$ for all $t \in I$ . Suppose that; - (i) $\mathcal{L}(z,t)$ is a locally absolutely continuous function in the interval I, and locally uniformly with respect to $\mathcal{U}_r$ . - (ii) $a_1(t)$ is a complex valued continuous function on I such that $a_1(t) \neq 0$ , $|a_1(t)| \to \infty$ for $t \to \infty$ and $$\left\{\frac{\mathcal{L}(z,t)}{a_1(t)}\right\}_{t\in I}$$ forms a normal family of functions in $\mathcal{U}_r$ . (iii) There exists an analytic function $p: \mathcal{U} \times I \to \mathbb{C}$ satisfying $\Re p(z,t) > 0$ for all $z \in \mathcal{U}, \ t \in I$ and (2.1) $$z \frac{\partial \mathcal{L}(z,t)}{\partial z} = p(z,t) \frac{\partial \mathcal{L}(z,t)}{\partial t} \quad (z \in \mathcal{U}_r, \ t \in I).$$ <span id="page-1-0"></span>Then, for each $t \in I$ , the function $\mathcal{L}(z,t)$ has an analytic and univalent extension to the whole disk $\mathcal{U}$ or the function $\mathcal{L}(z,t)$ is a Loewner chain. The equation (2.1) is called the generalized Loewner differential equation. The following strengthening of Theorem 2.1 leads to the method of constructing quasiconformal extension, and is based on the result due to Becker (see [3], [4] and also [5]).

Theorem 2.2. [3, 4, 5, Becker] Suppose that $\mathcal{L}(z,t)$ is a Loewner chain for which p(z,t), defined in (2.1), satisfies the condition $$p(z,t) \in U(k) := \left\{ w \in \mathbb{C} : \left| \frac{w-1}{w+1} \right| \le k \right\}$$ $$= \left\{ w \in \mathbb{C} : \left| w - \frac{1+k^2}{1-k^2} \right| \le \frac{2k}{1-k^2} \right\} \quad (0 \le k < 1)$$ for all $z \in \mathcal{U}$ and $t \in I$ . Then $\mathcal{L}(z,t)$ admits a continuous extension to $\overline{\mathcal{U}}$ for each $t \in I$ and the function $F(z,\overline{z})$ defined by $$F(z,\bar{z}) = \begin{cases} \mathcal{L}(z,0) & for \quad |z| < 1, \\ \mathcal{L}\left(\frac{z}{|z|},\log|z|\right) & for \quad |z| \ge 1, \end{cases}$$ is a k-quasiconformal extension of $\mathcal{L}(z,0)$ to $\mathbb{C}$ . Detailed information about Loewner chains and quasiconformal extension criterion can be found in [1], [2], [6], [7], [15], [23]. For a recent account of the theory we refer the reader to [10, 11, 12].

Theorem 3.1. Let $\alpha$ , c and s be complex numbers, that $c \notin [0, \infty)$ ; s = a + ib, a > 0, $b \in \mathbb{R}$ ; m > 0 and $f, g \in \mathcal{A}$ . If there exists a function h, analytic in $\mathcal{U}$ , and such that $h(0) = h_0$ , $h_0 \in \mathbb{C}$ , $h_0 \notin (-\infty, 0]$ , and the inequalities $$\left|\alpha - \frac{m}{2a}\right| < \frac{m}{2a},$$ $$\left| \frac{c}{h(z)} + \frac{m}{2\alpha} \right| < \frac{m}{2|\alpha|},$$ and <span id="page-2-3"></span> $$(3.3) \quad \left| \frac{-c\alpha}{ah(z)} |z|^{m/a} + \left( 1 - |z|^{m/a} \right) \left[ (\alpha - 1) \frac{zg'(z)}{g(z)} + 1 + \frac{zf''(z)}{f'(z)} + \frac{zh'(z)}{h(z)} \right] - \frac{m}{2a} \right| \le \frac{m}{2a}$$ hold true for all $z \in \mathcal{U}$ , then the function <span id="page-2-5"></span>(3.4) $$G_{\alpha}(z) = \left[\alpha \int_{0}^{z} g^{\alpha-1}(u) f'(u) du\right]^{1/\alpha}$$ is analytic and univalent in $\mathcal{U}$ , where the principal branch is intended.

Theorem 3.2. Let $f, g \in A$ . Let m > 0, the complex numbers $\alpha$ , c, s and the function h be as in Theorem 3.1. Moreover, suppose that the inequalities (3.1) and (3.2) are satisfied. If the inequality <span id="page-5-0"></span> $$\left| (\alpha - 1) \frac{zg'(z)}{g(z)} + 1 + \frac{zf''(z)}{f'(z)} + \frac{zh'(z)}{h(z)} - \frac{m}{2a} \right| \le \frac{m}{2a}$$ holds true for all $z \in \mathcal{U}$ , then the function $G_{\alpha}$ defined by (3.4) is analytic and univalent in $\mathcal{U}$ .

Theorem 3.5. Let $\alpha$ , c and s be complex numbers, that $c \notin [0, \infty)$ ; s = a + ib, $a \ge 1$ , $b \in \mathbb{R}$ ; m > 0 and $f, g \in \mathcal{A}$ . Let the function h be as in Theorem 3.1. Moreover, suppose that the inequalities (3.1) and (3.2) are satisfied. If the inequality <span id="page-6-0"></span> $$(3.16) \qquad \left| \frac{-c\alpha}{ah(z)} |z|^m + (1 - |z|^m) \left[ (\alpha - 1) \frac{zg'(z)}{g(z)} + 1 + \frac{zf''(z)}{f'(z)} + \frac{zh'(z)}{h(z)} \right] - \frac{m}{2a} \right| \le \frac{m}{2a}$$ holds true for all $z \in \mathcal{U}$ , then the function $G_{\alpha}(z)$ defined by (3.4) is analytic and univalent in $\mathcal{U}$ .

Theorem 4.1. Let $\alpha$ , c and s be complex numbers, that $c \notin [0, \infty)$ ; s = a + ib, a > 0, $b \in \mathbb{R}$ ; m > 0; $k \in [0, 1)$ and let $f, g \in \mathcal{A}$ . If there exists a function h, analytic in $\mathcal{U}$ , such that $h(0) = h_0$ , $h_0 \in \mathbb{C}$ , $h_0 \notin (-\infty, 0]$ and the inequalities <span id="page-7-1"></span> $$\left|\alpha - \frac{m}{2a}\right| < \frac{m}{2a}$$ $$\left| \frac{c\alpha}{h(z)} + \frac{m}{2} \right| < k \frac{m}{2}$$ and <span id="page-7-2"></span> $$(4.2) \quad \left| \frac{-c\alpha}{ah(z)} |z|^{m/a} + \left( 1 - |z|^{m/a} \right) \left[ (\alpha - 1) \frac{zg'(z)}{g(z)} + 1 + \frac{zf''(z)}{f'(z)} + \frac{zh'(z)}{h(z)} \right] - \frac{m}{2a} \right| \le k \frac{m}{2a}$$ hold true for all $z \in \mathcal{U}$ , then the function $G_{\alpha}(z)$ given by (3.4) has an K-quasiconformal extension to $\mathbb{C}$ , where (4.3) $$K = \begin{cases} k & for \quad s = 1, \\ \frac{|s-1|^2 + k|\bar{s}^2 - 1|}{|\bar{s}^2 - 1| + k|s - 1|^2} & for \quad s \neq 1. \end{cases}$$

Theorem 4.2. Let $\alpha > 0$ , and $f, g \in \mathcal{A}$ . If $$z^{1-\alpha}g(z)^{\alpha-1}f'(z) \in U(k)$$ for all $z \in \mathcal{U}$ , then the function $G_{\alpha}(z)$ can be extended to a k-quasiconformal automorphism of $\mathbb{C}$ . Proof. Set $$\mathcal{L}(z,t) = \left(\alpha \int_{0}^{z} g^{\alpha-1}(u)f'(u)du + (e^{\alpha t} - 1)z^{\alpha}\right)^{1/\alpha}.$$ An easy computation shows $$p(z,t) = \frac{1}{e^{\alpha t}} \left( z^{1-\alpha} g(z)^{\alpha-1} f'(z) \right) + \left( 1 - \frac{1}{e^{\alpha t}} \right),$$ and the assertion follows by the same methods as in Theorem 4.1, applying Theorems 2.1 and 2.2. $\Box$ In the same manner, by definition of the suitable Loewner chain, several univalence criterion may by found. For example, the condition $$\frac{zG'_{\alpha}(z)}{G_{\alpha}(z)} \in U(k) \quad (\alpha \in \mathbb{C}),$$ which is based on the integral operator $G_{\alpha}(z)$ , is given by the Loewner chain $$\mathcal{L}(z,t) = e^t G_{\alpha}(z).$$ This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge, Faculty of Mathematics and Natural Sciences, University of Rzeszow.