On the range of holomorphic functions: from Landau to Picard’s theorems

Abstract

[en] The range of a holomorphic function is a classical topic in complex variable. Throughout this project, we will give several results on the size of this range. Among them we mention: Landau's Injective Theorem, Landau's Covering Theorem, Bloch's Theorem and Picard's Theorems.

The first two results analyse the uniqueness in Schwarz's Lemma giving a precise estimate on the size of both the biggest disc covered by the function and the biggest disc where the function is injective, in terms of $\left|f^{\prime}(0)\right|$. Bloch's Theorem is also a covering result with fewer hypotheses on the function, and it is a key tool in the proof of Picard's Theorem. Finally, we prove both Picard's Little and Great theorems. The first states that any entire function that omits at least two values is constant, and the second one, which holds for meromorphic functions, can be viewed as a generalisation of the Casorati-Weierstraß' theorem. Finally, along the third and fourth chapters, we will see which consequences derive from these two classical results.