[16] Sur l’équation différentielle de M. Löwner

Abstract

Twenty years after its appearance in 1923, Loewner’s method was well established as a powerful device for solving extremal problems. For instance, Grunsky had applied it in 1934 to find the radius of starlikeness for univalent functions of class S. Goluzin had applied it in 1936 to obtain the sharp rotation theorem. Loewner (Math. Ann., 89, 103–121, 1923) had originally used his method to prove |a ₃|≤ 3. (See Goluzin, Geometric Theory of Functions of a Complex Variable, Nauka, Moscow, 1966 or Duren, Univalent Functions, 1983 for further information). In his short note (Schiffer, C. R. Acad. Sci. Paris, 221, 369–371, 1945) of 1945, Schiffer developed a clever idea for combining Loewner’s method with the variational method in order to mount a stronger attack on the Bieberbach conjecture. Schiffer had applied his method of boundary variation (Schiffer, Proc. London Math. Soc., 44(2), 432–449, 1938) to show in Schiffer, (Proc. London Math. Soc. 44(2), 450–452, 1938) that an extremal function for the coefficient problem max _f∈S Re{a _n } maps the unit disk $\mathbb{D}$ conformally onto the whole plane minus a finite number of unbranched analytic arcs that are trajectories of a quadratic differential: $(1/{w}^{3})P_{n}(1/w)\,d{w}^{2} < 0$, where P _n is a certain polynomial of degree n − 2. (See also Duren, Univalent Functions, 1983. Observe that our present notation differs from that in Schiffer’s paper, where P _n denotes a polynomial of degree n.) In particular, if w = w(s) is a sufficiently smooth (local) parametric representation of an omitted arc, then Peter L.

$$\displaystyle{ \text{Im}\left \{w^{\prime}{(s)}^{2}w{(s)}^{-3}P_{ n}(1/w(s))\right \} = 0\,. }$$

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