Abstract
In this paper, we find conditions on the coefficients {b(k)}(k=1)(n) such that the corresponding trigonometric (cosine and sine) sums given respectively by Sigma(n)(k=1) b(k) sin k theta > 0 and Sigma(n)(k=1) b(k) cos k theta > 0 for all n epsilon N are positive. Using these results, we find that the functions f that are in the class of analytic functions A are starlike of certain order in the unit disc D by means of conditions on the Taylor coefficients of f. As an application, we also find conditions such that the Cesaro means of order beta of f (z) are close-to-convex and starlike in D. (C) 2011 Elsevier Ltd. All rights reserved.