  The Markov-Bernstein type inequalities between the norms of functions and of
their derivatives are analysed for complex exponential polynomials. We
establish a relation between the sharp constants in those inequalities and the
stability problem for linear switching systems. In particular, the maximal
discretization step is estimated. We prove the monotonicity of the sharp
constants with respect to the exponents, provided those exponents are real.
This gives asymptotically tight uniform bounds and the general form of the
extremal polynomial. The case of complex exponent is left as an open problem.
