Let / be a function analytic in the unit disc, properly normalized, with bounded boundary rotation. There exists a Stieltjes integral representation for 1 +zf"(z)jf'(z). From this representation, and in view of a known variational formula for functions of positive real part, a variational formula is derived for functions of the form q(z) = l +zf"(z)jf'(z). This formula is for functions of arbit...

In this paper we introduce certain analytic functions of boundary rotation bounded by k1r which are of Caratheodory origin. With them we study two classes of analytic and univalent functions in the unit disk E = {z E C Izi < I} , which are also of bounded boundary rotation.

The paper studies the third problem for the Laplace equation on a cracked bounded planar domain with multiply connected Lipschitz boundary and boundary conditions from Lp. It is shown that, for 1 < p ≤ 2, there is a unique solution of the problem. This solution is constructed for a domain, which boundary is formed by curves with bounded rotation.