Let f ∊ C[0, 1] and let the Bn(f, q; x) be generalized Bernstein polynomials based on the q-integers that were introduced by Phillips. We prove that if f is r-monotone, then Bn(f, q; x) is r-monotone, generalizing well-known results when q = 1 and the results when r = 1 and r = 2 by Goodman et al. We also prove a sufficient condition for a continuous function to be r-monotone.

We study various Dirichlet series of the form ∑n≥1f(πnα)/ns, where α is an irrational number and f(x) is a trigonometric function like cot(x), 1/sin(x) or 1/sin2(x). The convergence is slow and strongly depends on the Diophantine properties of α. We provide necessary and sufficient convergence conditions using the continued fraction of α. We also show that any one of our series is equal to a related series, which converges much faster, defined in term of iterations of the continued fraction operator α↦{1/α}.

We find approximate solutions (chord–arc curves) for the system of equations of geodesics in Ω∩ℍ for every Denjoy domain Ω, with respect to both the Poincaré and the quasi-hyperbolic metrics. We also prove that these chord–arc curves are uniformly close to the geodesics. As an application of these results, we obtain good estimates for the lengths of simple closed geodesics in any Denjoy domain, and we improve the characterization in a 1999 work by Alvarez et al. on Denjoy domains satisfying the linear isoperimetric inequality.