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:|z|< 1} such that |f(z)|< 1 for all z is an element of D . If f is an element of B of the form f(z)= Sigma(infinity)(n=0) a(n)z(n) , then Sigma(infinity)(n=0) |a(n)z(n)| <= 1 for |z| = r <= 1/3 and 1/3 cannot be improved. This inequality is called Bohr inequality and the quantity 1/3 is called Bohr radius. If f is an element of B of the form f(z)= Sigma(infinity)(n=0)a(n)z(n) , then |Sigma(N)(n=0) a(n)z(n)| < 1 for |z| < 1/2 and the radius 1/2 is the best possible for the class B . This inequality is called Bohr-Rogosinski inequality and the corresponding radius is called Bohr-Rogosinski radius. Let H be the class of all complex-valued harmonic functions f=h+(g) over bar defined on the unit disk D , where h and g are analytic in D with the normalization h(0)=h '(0)-1=0 and g(0)=0 . Let H-0={f=h+(g) over bar is an element of H:g '(0)=0}. For alpha >= 0 and 0 <=beta < 1 , let W-H(0)(alpha,beta)={f=h+(g) over bar is an element of H0:Re(h '(z)+alpha zh ''(z)-beta)>|g '(z)+alpha zg ''(z)|,z is an element of D} be a class of close-to-convex harmonic mappings in D . In this paper, we prove the sharp Bohr-Rogosinski radius for the class W-H(0)(alpha,beta) .