Inverse function theorem on the class of holomorphic self-maps of a disc with two fixed points

Abstract

In the present paper we solve the problem on the sharp domain of invertibility on the class of holomorphic self-maps of the unit disc with two (interior and boundary) fixed points and a constraint on the angular derivative at the boundary fixed point. The interest in such extremal problems stems primarily from Bloch’s famous theorem [1] to the effect that any function f holomorphic in the unit disc D = {z ∈ C : |z| < 1} is invertible in some disc of radius R|f ′(0)|, where R is an absolute constant. The search for the sharp upper bound B of such R’s (known as the Bloch constant) is one of the most important (and still unsolved) problems of geometric function theory. The near-best lower estimate B ⩾ √ 3/4 is due to Ahlfors [2]. Considerably later, Bonk [3] proved with the help of his distortion theorem on the Bloch class that Ahlfors’ estimate is not sharp, that is, B > √ 3/4. A little later, Chen and Gauthier [4] showed that B > √ 3/4 + 2 · 10−4 by slightly improving the technical details of Bonk’s proof. In our opinion, Landau’s approach might be capable of delivering new lower bounds for the Bloch constant. Considering the class of bounded holomorphic maps f of the disc D with interior fixed point z = 0 and such that f ′(0) = 1, Landau [5] proved the existence of a common disc of univalence on this class and found its precise radius. Moreover, he discovered that there is a disc in which all functions from this class are invertible, and he also determined the precise radius of this disc. Using these results, Landau gave one of the first estimates of the Bloch constant. At the same time, the domain of invertibility of each function in the class studied by Landau is much broader than the common disc of invertibility. This suggest the natural problem of finding sharp domains of univalence and invertibility on subclasses of this class. In a certain sense, as such subclasses it is natural to study the classes of holomorphic self-maps of the unit disc D with several fixed points (see [6]), which have important applications. We let B denote the class of holomorphic self-maps of D. Putting B[0] = {f ∈ B : f(0) = 0}, we can write Landau’s results as follows: if f ∈ B[0] and |f ′(0)| ⩾ 1/M with M > 1, then f is univalent in Z = {z ∈ D : |z| < M− √ M2 − 1 } and invertible in W = {w ∈ D : |w| < (M − √ M2 − 1 )}. Moreover, in place of Z and W one can take neither discs of larger radius nor any broader domains. The proof of this result is based on the following inequality (see [5]): if f ∈ B[0] and if a, b ∈ D with a ̸= b are such that f(a) = f(b) = c, then |c| ⩽ |a| |b|. On the class B{1} = {f ∈ B : ∠ limz→1 f(z) = 1} Becker and Pommerenke [7] obtained an inequality analogous to Landau’s in a certain sense. They showed that