The Schwarzian derivative and multivalence

Theorem 2.1. Let $y_i$ , $i = 1, 2, \dots, n$ , be linearly independent solutions of (2.1), let $$(2.3) f_1 = \frac{y_1}{y_n}, \dots, f_{n-1} = \frac{y_{n-1}}{y_n}$$ and let $W_i$ be the determinant defined by $$W_i = \begin{array}{ccccc} f_1' & f_2' & \cdots & f_{n-1}' \ & \ddots & & & & \ f_1^{(i-1)} & f_2^{(i-1)} & \cdots & f_{n-1}^{(i-1)} \ f_1^{(i+1)} & f_2^{(i+1)} & \cdots & f_{n-1}^{(i+1)} \ & & \ddots & & \ f_1^{(n)} & f_2^{(n)} & \cdots & f_{n-1}^{(n)} \ \end{array} ,$$ $i = 1, 2, \dots, n$ . Then we have $$(2.4) \quad p_i = \frac{1}{W_n \sqrt[n]{\overline{W_n}}} \left[ \sum_{j=0}^{n-i} (-1)^{2n-j} (1-\delta_{nj}) \binom{n-j}{n-j-i} W_{n-j} (\sqrt[n]{\overline{W_n}})^{(n-j-i)} \right],$$ $i = 0, 1, \dots, n-2$ , where $\delta_{nn} = 1$ and $\delta_{nj} = 0$ otherwise. Conversely, the general solution $(f_1, f_2, \dots, f_{n-1})$ of the system (2.4) of differential equations is of the form (2.3).

Theorem 3.1. Let $y_1, y_2, \dots, y_n$ be linearly independent solutions of (2.1), and let $f_i = y_i/y_n$ , $i = 1, 2, \dots, n-1$ . Then the differential equation (2.1) is disconjugate in D if and only if every nontrivial linear combination of $f_1, f_2, \dots, f_{n-1}$ is (n-1)-valent in D, i.e., it does not take on any one value more than n-1 times in D.

Lemma 3.1. Let y be analytic in a region R. If $y(a_i) = 0$ , $a_i \in R$ , $i = 1, 2, \dots, n$ , then $$(3.1) y^{(k)}(z) = \sum_{j=1}^{k+1} {k \choose j-1} P_{n-j}^{(k+1-j)}(z) I_j(z) (a_j-z)^{-j+1},$$ $k = 0, 1, \dots, n - 1, where$ $$I_n(z)=\int_{a_n}^z (a_n-\zeta)^{n-1}y^{(n)}(\zeta)d\zeta$$ , $I_j(z)=\int_{a_j}^z \frac{(a_j-\zeta)^{j-1}}{(a_{j+1}-\zeta)^{j+1}}I_{j+1}(\zeta)d\zeta$ , $j=1,2,\cdots,n-1$ , and $$P_{n-j}(z) = \prod_{i=j+1}^{n} (a_i - z)$$ .

Theorem 3.2. Let $p_0, \dots, p_{n-1}$ be analytic in the unit disk $D = \{z: |z| < 1\}$ . If $$(3.2) \quad \sum_{k=1}^{n-1} \frac{(1+|z|)^{n-k}}{(n-k)!} |p_k(z)| + \frac{(1-|z|)(1+|z|)^{n-1}}{n!} |p_0(z)| \leq 1,$$ then the differential equation $$(3.3) y^{(n)} + p_{n-1}(z)y^{(n-1)} + \cdots + p_0(z)y = 0$$ is disconjugate in D. 722 W. J. KIM

Theorem 3.3. Let $f_1, f_2, \dots, f_{n-1}$ be analytic in the unit disk $D = \{z: |z| < 1\}$ . Define $p_0, p_1, \dots, p_{n-2}$ as in (2.4). If $\det(f_j^{(i)})_{i,j=1}^{n-1}$ does not vanish in D, and if $$\begin{align} \sum_{k=1}^{n-2} \frac{(1\,+\,|z\,|)^{n-k}}{(n\,-\,k)!}\,|\,p_k(z)\,| \ &+\frac{1}{n!}(1\,-\,|z\,|)(1\,+\,|z\,|)^{n-1}\,|\,p_{\scriptscriptstyle 0}(z)\,|\,\leqq 1,\,z\,{\in}\,D \;, \end{gathered}$$ then every nontrivial linear combination of $f_1, f_2, \dots, f_{n-1}$ is (n-1)-valent in D.