On the positivity of certain trigonometric sums and their applications

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 @?k=1nb"[email protected]>0 and @?k=1nb"[email protected]>0 for all [email protected]?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 @b of f(z) are close-to-convex and starlike in D.

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