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In this paper, we find conditions on the coefficients [Formula: see text] such that the corresponding trigonometric (cosine and sine) sums given respectively by ∑k=1nb_ksinkθ>0 and ∑k=1nb_kcoskθ>0 for all n∈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 Cesáro means of order β of f(z) are close-to-convex and starlike in D.