Abstract
Starting out from a question posed by T. Erdélyi and J.
Szabados, we consider Schur-type inequalities for the classes of
complex algebraic polynomials having no zeros within the unit
disk D. The class of polynomials with no zeros in D—also known as
Bernstein or Lorentz class—was studied in detail earlier. For
real polynomials utilizing the Bernstein-Lorentz representation as
convex combinations of fundamental polynomials
(1−x)k(1+x)n−k, G. Lorentz, T. Erdélyi, and J. Szabados
proved a number of improved versions of Schur- (and also
Bernstein- and Markov-) type inequalities. Here we investigate the similar questions for complex polynomials. For complex polynomials, the above convex representation is not
available. Even worse, the set of complex polynomials, having no
zeros in the unit disk, does not form a convex set. Therefore, a
possible proof must go along different lines. In fact, such a
direct argument was asked for by Erdélyi and Szabados already
for the real case. The sharp forms of the Bernstein- and Markov-type inequalities
are known, and the right factors are worse for complex
coefficients than for real ones. However, here it turns out that
Schur-type inequalities hold unchanged even for complex
polynomials and for all monotonic, continuous weight functions. As
a consequence, it becomes possible to deduce the corresponding
Markov inequality from the known Bernstein inequality and the new
Schur-type inequality with logarithmic weight.