The Bohr-Rogosinski Radius for a Certain Class of Close-to-Convex Harmonic Mappings

Let B be the class of analytic functions f in the unit disk D := {z is an element of C : vertical bar z vertical bar < 1} such that vertical bar f (z) vertical bar< 1 for all z. D. If f. B is of the form f (z) = Sigma(infinity)(n=0) a(n) z(n), then vertical bar Sigma(N)(n=0) an zn vertical bar < 1 holds for |z| < 1/2 and the radius 1/2 is best possible for the class B. This inequality is called the Rogosinski inequality and the corresponding radius is called the Rogosinski radius. Let H be the class of harmonic functions f = h + (g) over bar in the unit disk D, where h and g are analytic in D. Let P-H(0) (a) = {f = h + (g) over bar is an element of H : Re(h' (z) - alpha) > |g (z)| with 0 <= alpha < 1, g' (0) = 0, z. D} be the subclass of close-to-convex harmonic mappings. In this paper, in view of the Euclidean distance, we obtain the sharp Bohr-Rogosinski radius in terms of area measure Sr, Jacobian J f ( z) of the functions in the class P0H (a).