ON THE THEORY OF SCHLICHT FUNCTIONS

Abstract

Let us denote by $k$ a certain positive integer.If $ f(z)=z+c_{1}z^{k+1}+c_{2}z^{2k+1}+\ldots.+c_{n}z^{nk+1}+\ldots$ .is regular and schlicht in the unit circle, then we may say, for con- venience, that $f(z)$ is a function of the class $k$ .If $f(z)$ is a function of the class $k$ and maps the unit circle on a starshaped domain with centre at the origin, then we may say, for brevity, that $f(z)$ is a starshaped function of the class $k$ .Similarly, if $f(z)$ is a function of the class $k$ and maps the unit circle on a convex domain, we may call $f(z)$ a convex function of the class $k$ .Our main object in this paper is to state some theorems concerning a starshaped resp.convex function of the class $k^{(1)}$ , which are known when $k=1$ or 2. First we will obtain some results on the coefficients, using an easy lemma.Next, some extensions of STROHHACKER'S theorems will be mentioned.Applying the above results, we can extend SZEG\"o'S theorem on the polynomial sections of a starshaped resp.convex function.It is well known that under the condition that $f(z)$ is regular in $D$ and that $f^{\prime}(z)$ never vanishes there, we cannot necessarily assert $f(z)$ to be schlicht in $D$ .In \S V, imposing a further condition on (1) Recently Mr. CHEN has obtained some results concerning a (schlicht) function of the class $k$ .See