The hermitian u-invariants of a central simple algebra with involution are studied. In this context, a new technique is obtained to give bounds for the behavior of these invariants under a quadratic field extension. This is applied to obtain bounds in terms of the index and the $u$-invariant of the base field.

We wish to express our gratitude to Nicolas Garrel, Archita Mondal, and Anne Quequiner-Mathieu for inspiring discussions and\break many helpful comments and suggestions. We would also like to thank the referee for their careful reading and valuable feedback. This work was supported by the Fonds Wetenschappelijk Onderzoek – Vlaanderen in the FWO Odysseus Programme (project G0E6114N, "Explicit Methods in Quadratic Form Theory"), by the FWO-Tournesol programme (project VS05018N), by the Fondazione Cariverona in the programme Ricerca Scientifica di Eccellenza 2018 (project "Reducing complexity in algebra, logic, combinatorics -- REDCOM"), by the 2020 PRIN (project "Derived and underived algebraic stacks and applications") from MIUR, and by research funds from Scuola Normale Superiore.