On the u-invariant of hermitian forms

Let K be a field of characteristic not 2 and A a central simple algebra with an involution σ. A result of Mahmoudi provides an upper bound for the u-invariants of hermitian forms and skew-hermitian forms over (A, σ) in terms of the u-invariant of K. In this paper we give a different upper bound when A is a tensor product of quaternion algebras and σ is a the tensor product of canonical involutions. We also show that our bounds are sharper than those of Mahmoudi.