Inequalities for meromorphic functions with prescribed poles

The extremal problems for functions of complex variables, as well as approaches for obtaining classical inequalities on the base of various methods of the geometric function theory, are known for various norms and for many classes of functions such as rational functions with various constraints and for various domains in the complex plane. It is important to mention that different types of Bernstein-type inequalities appeared in the literature in more generalized forms in which the underlying polynomial was replaced by a more general class of functions. One such generalization is the passage from polynomials to rational functions. In this paper, we prove some inequalities for meromorphic functions with prescribed poles and restricted zeros. These results not only generalize some Bernstein-type inequalities for rational functions, but also improve and generalize some known polynomial inequalities. These inequalities have their own importance in the approximation theory.