The coefficient regions of starlike functions

Lemma 1. $C_n^*$ is a closed, convex set. Received June 24, 1957. The work reported on here was done while the writer held a National Science Foundation post doctoral Fellowship. The writer wishes to thank Professor M. Schiffer for many helpful conversations during the course of this work.

Lemma 2. For $1 \le k \le m$ , the polynomial $R_k(z)$ is of the form $$R_k(z) = \alpha_{0,k} + \alpha_{1,k}z + \dots + \alpha_{n-k-1,k}z^{n-k-1} + \dots + \beta_{n-k-1,k}^z^{n-1} + \dots + \beta_{n-k-1,k}^z^{n-1} + \dots + \beta_{n-k-2,k}^z^n + \dots + \beta_{n-k-2,k}^z^{n-k-2},$$ with $$\alpha_{\mu,k} = \lambda_n A_{\mu+1,+k} \lambda_{n-1} A_{\mu,k} + \dots + \lambda_{n-\mu} A_{1,k} ,$$ $$\beta_{\mu,k} = \lambda_n B_{\mu+1,k} + \lambda_{n-1} B_{\mu,k} + \dots + \lambda_{n-\mu} B_{1,k} .$$ Here, each $A_{j,k}$ and each $B_{j,k}$ is a polynomial in the $a_{\gamma}$ and their conjugates (independent of the $\lambda_{\gamma}$ ), and the $A_{j,k}$ and $B_{j,k}$ satisfy the recursion relations (7) $$\begin{cases} A_{j,\nu+1} = B_{1,\nu} A_{j+1,\nu} - A_{1,\nu} B_{j+1,\nu}, \\ B_{j,\nu+1} = B_{1,\nu}^ B_{j,\nu} - A_{1,\nu}^ A_{j,\nu}. \end{cases}$$

Lemma 3. For each $j, k, 0 \le j \le n-k-1, 1 \le k \le m$ , the $A_{j,k}$ and $B_{j,k}$ of Lemma 2 satisfy the following: - (i) $A_{j,k}$ is a polynomial in $a_2, a_3, \dots, a_{j+k}$ and $a_2^, a_3^, \dots, a_{k-1}^*$ . - (ii) $B_{j,k}$ is a polynomial in $a_2,a_3,\cdots,a_{j+k-1}$ and $a_2^,a_3^,\cdots,a_k^*$ . - (iii) $B_{1,k}$ is real for any choice of $a_2, a_3, \dots, a_k$ . (iv) $$A_{j,k} = (j+k-1)B_{1,1}B_{1,2} \cdots B_{1,k-1}a_{j+k} - A_{1,1}B_{1,2}B_{1,3} \cdots B_{1,k-1}B_{j+k-1,1}$$ $-A_{1,2}B_{1,3}B_{1,4} \cdots B_{1,k-1}B_{j+k-2,2} - \cdots - A_{1,k-2}B_{1,k-1}B_{j+2,k-2}$ $-A_{1,k-1}B_{j+1,k-1}.$ (v) For any $$\nu$$ , $1 \leq \nu < k$ $$B_{1,k} = B_{1,\nu}^2 B_{1,\nu+1} B_{1,\nu+2} \cdots B_{1,k-1} - |A_{1,k-1}|^2 - B_{1,k-1} |A_{1,k-2}|^2$$ $$-B_{1,k-1}B_{1,k-2}|A_{1,k-3}|^2 - \cdots$$ $-B_{1,k-1}B_{1,k-2}\cdots B_{1,k+1}|A_{1,k}|^2$ .

Theorem 1. Let $(a_2, a_3, \dots, a_{n-1}) \in V_{n-1}$ . If $(a_2, a_3, \dots, a_{n-1})$ is an interior point of $V_{n-1}$ then $C_n^(a_2, \dots, a_{n-1})$ is a circular disc determined by $|A_{1,n-1}| = B_{1,n-1}$ ; furthermore $|A_{1,k}| < B_{1,k}$ for k < n-1. If $(a_2, \dots, a_{n-1})$ is a boundary point of $V_{n-1}$ then $C_n^*(a_2, \dots, a_{n-1})$ consists of a single point.