Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann-Hilbert-Poincaré problem

I. vekuas integral representations of holomorphic functions,whose m-th derivative (m ≥ 0) is hölder-continuous in a closed domain bounded by the lyapunov curve,are generalized for analytic functions whose m-th derivative is representable by a cauchy type integral whose density is from variable exponent lebesgue space l p(̇)(γ;ω) with power weight. an integration curve is taken from a wide class of piecewise-smooth curves admitting cusp points for certain p and ω. this makes it possible to obtain analogues of i. vekuas results to the riemann-hilbert-poincaré problem under new general assumptions about the desired and the given elements of the problem. it is established that the solvability essentially depends on the geometry of a boundary,a weight function ω(t) and a function p (t). copyright © 2011 hindawi publishing corporation.