  We introduce a condition on accretive matrix functions, called
$p$-ellipticity, and discuss its applications to the $L^p$ theory of elliptic
PDE with complex coefficients. Our examples are:
  (i) generalized convexity of power functions (Bellman functions),
  (ii) dimension-free bilinear embeddings,
  (iii) $L^p$-contractivity of semigroups and
  (iv) holomorphic functional calculus.
  Recent work by Dindo\v{s} and Pipher (arXiv:1612.01568v3) established close
ties between $p$-ellipticity and
  (v) regularity theory of elliptic PDE with complex coefficients.
  The $p$-ellipticity condition arises from studying uniform positivity of a
quadratic form associated with the matrix in question on one hand, and the
Hessian of a power function on the other. Our results regarding contractivity
extend earlier theorems by Cialdea and Maz'ya.
