Integral estimates of derivatives of rational functions in Hölder domains

Given a bounded rational function of degree $n$ in a Hölder domain in the complex plane, it is shown that the area integral of the modulus of its derivative is bounded by a quantity of order $\sqrt{\log n}$. The obtained inequality improves a classical result of E.P. Dolzhenko (1966), as well as some of our recent results. Examples are constructed illustrating the influence of the length of the boundary on the behavior of area integrals of the moduli of the derivatives of bounded rational functions.