Markov–Bernstein–Tur´ an type inequalities for complex polynomials T. Erd´ elyi and J. Szabados* 1. Introduction We will be concerned with generalizations of the following classical polynomial in- equalities ||p ′ || I ≤ n 2 ||p|| I (1.1) (Markov inequality), and ||p ′ || D ≤ n||p|| D (1.2) (Bernstein inequality), where I := [−1, 1], D is the closed unit disk, p ∈P c n (=the space of polynomials of degree at most n with complex coefficients), and || · || means supremum norm over the set specified. There is a striking difference beteween n and n 2 (the so- called Markov-factors) in the above inequalities, due to the difference between the domains considered. Our purpose in this paper is to create a transition between these Markov- factors by considering ellipses whose limit cases are I and D. We shall also consider Tur´ an type inequalities in these domains, i.e. we intend to give lower estimates for the Markov factors when the roots of the polynomials are constrained to these domains. Apart from constants, these inequalities turn out to be sharp. We shall also consider these problems for diamond shaped domains, but the results for such domains will be less complete. 2. Markov type inequalities The most natural way of seeking connection between the inequalities (1.1) and (1.2) is to introduce the ellipse E ε := {z = x + iy : ε 2 x 2 + y 2 ≤ ε 2 } , 0 ≤ ε ≤ 1 , in the complex plane, and establish an inequality there. (ε = 0 and ε = 1 correspond to (1.1) and (1.2), respectively.) Theorem 1. There exists an absolute constant C> 0 such that 1 C min  n ε ,n 2  ≤ sup p∈P c n ||p ′ || E ε ||p|| E ε ≤ min C  n ε ,n 2  , 0 ≤ ε ≤ 1 . (2.1) * Research of this author supported by OTKA No. 034531 and 037299. 1