A Schwarz-Jack Lemma for bi-circular symmetric domains

The Schwarz lemma is one of the cornerstones of geometric function theory. Jack’s lemma is another influential result in the field, though it is somewhat less well known. In this work, we introduce a new result, the Schwarz–Jack lemma, which is strongly inspired by both of these classical lemmas. The main requirement of the lemma, known as the Jack condition, can be satisfied in several different ways, some of which we discuss. This result emerged from our study of the celebrated Crouzeix conjecture, which asserts that the numerical range of any \(n \times n\) matrix is a 2-spectral set. While the conjecture has been established for the case \(n = 2\), it remains open for \(n \geq 3\). Existing proofs for \(n = 2\) are highly technical and rely heavily on explicit formulas for conformal mappings from an ellipse onto the unit disk. The broader aim of this work is twofold: first, to clarify the intrinsic properties of the conformal mappings involved, thereby reducing reliance on explicit formulas; and second, to provide a conceptual framework that may allow for further generalizations. The Schwarz–Jack lemma plays a central role in achieving these goals, in particular for the first one. It is used to derive a general result on conformal mappings over bi-oval domains, offering an abstract perspective that both underpins the known proofs of the Crouzeix theorem for \(n = 2\) and potentially points toward new approaches in higher dimensions.

Joint work with A. Moucha, R. O’Loughlin, O. Roth, and T. Ransford.