Optimal estimates for derivatives of analytic functions and solutions to Laplace, Lam\'e and Stokes equations

Two types of optimal estimates for derivatives of analytic functions
with bounded real part are considered. The first of them is a pointwise
inequality for derivatives of analytic functions in the complement
of a convex closed domain in ${\mathbb C}$. The second type of inequalities
is a limit relation for derivatives of analytic functions in an arbitrary proper
subdomain of ${\mathbb C}$. Optimal estimates for derivatives of a vector
field with bounded harmonic components as well as optimal estimates for the
divergence of an elastic displacement field and pressure in a fluid in
subdomains of ${\mathbb R}^n$ are discussed.