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On the positivity of certain trigonometric sums and their applications. (English) Zbl 1236.42002
Summary: We find conditions on the coefficients {b k } k=1 n such that the corresponding trigonometric (cosine and sine) sums given respectively by k=1 n b k sinkθ>0 and k=1 n b k coskθ>0 for all 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.
MSC:
42A32Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
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