  For a class of volume preserving partially hyperbolic diffeomorphisms (or
non-uniformly Anosov) $f:\mathbb{T}^d \rightarrow \mathbb{T}^d$ homotopic to
linear Anosov automorphism, we show that the sum of the positive (negative)
Lyapunov exponents of $f$ is bounded above (resp. below) by the sum of the
positive (resp. negative) Lyapunov exponents of its linearization. We show this
for some classes of derived from Anosov (DA) and non-uniformly hyperbolic
systems with dominated splitting, in particular for examples described by
Bonatti and Viana.
