  Controlling the monotonicity and growth of Leray--Lions' operators including
the $p$-Laplacian plays a fundamental role in the theory of existence and
regularity of solutions to second order nonlinear PDE. We collect, correct, and
supply known estimates including the discussion on the constants. Moreover, we
provide a comprehensive treatment of related results for operators with Orlicz
growth. We pay special attention to exposition of the proofs and the use of
elementary arguments.
