  Univalent functions are complex, analytic (holomorphic) and injective
functions that have been widely discussed in complex analysis. It was recently
proposed that the stringent constraints that univalence imposes on the growth
of functions combined with sufficient analyticity conditions could be used to
derive rigorous lower and upper bounds on hydrodynamic dispersion relation,
i.e., on all terms appearing in their convergent series representations. The
results are exact bounds on physical quantities such as the diffusivity and the
speed of sound. The purpose of this paper is to further explore these ideas,
investigate them in concrete holographic examples, and work towards a better
intuitive understanding of the role of univalence in physics. More concretely,
we study diffusive and sound modes in a family of holographic axion models and
offer a set of observations, arguments and tests that support the applicability
of univalence methods for bounding physical observables described in terms of
effective field theories. Our work provides insight into expected `typical'
regions of univalence, comparisons between the tightness of bounds and the
corresponding exact values of certain quantities characterizing transport,
tests of relations between diffusion and bounds that involve chaotic
pole-skipping, as well as tests of a condition that implies the conformal bound
on the speed of sound and a complementary condition that implies its violation.
