Rigidity of Lyapunov exponents for derived from Anosov diffeomorphisms

For a class of volume preserving partially hyperbolic diffeomorphisms (or non-uniformly Anosov) $f\colon {\T}^d\rightarrow{\T}^d$ 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 C. Bonatti and Viana. The results in this paper address a flexibility program by J. Bochi, A. Katok and F. Rodriguez Hertz.