  This note contains some observations on primary matrix functions and
different notions of monotonicity with relevance towards constitutive relations
in nonlinear elasticity. Focussing on primary matrix functions on the set of
symmetric matrices, we discuss and compare different criteria for monotonicity.
The demonstrated results are particularly applicable to computations involving
the true-stress-true-strain monotonicity condition, a constitutive inequality
recently introduced in an Arch. Appl. Mech. article by C.S. Jog and K.D. Patil.
We also clarify a statement by Jog and Patil from the same article which could
be misinterpreted.
