On the positivity of certain trigonometric sums and their applications

In this paper, we find conditions on the coefficients {bk}k=1n such that the corresponding trigonometric (cosine and sine) sums given respectively by ∑k=1nbksinkθ>0 and ∑k=1nbkcoskθ>0 for all n∈Nn∈N are positive. Using these results, we find that the functions ff that are in the class of analytic functions AA are starlike of certain order in the unit disc DD by means of conditions on the Taylor coefficients of ff. As an application, we also find conditions such that the Cesáro means of order ββ of f(z)f(z) are close-to-convex and starlike in DD.