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Zbl 1236.42002
Mondal, Saiful R.; Swaminathan, A.
On the positivity of certain trigonometric sums and their applications.
(English)
[J] Comput. Math. Appl. 62, No. 10, 3871-3883 (2011). ISSN 0898-1221

Summary: We find conditions on the coefficients $\{b_k\}_{k=1}^n$ such that the corresponding trigonometric (cosine and sine) sums given respectively by $\sum_{k=1}^nb_k\sin k\theta>0$ and $\sum_{k=1}^n b_k\cos k\theta>0$ for all $n\in \bbfN$ are positive. Using these results, we find that the functions $f$ that are in the class of analytic functions $\cal A$ are starlike of certain order in the unit disc $\bbfD$ by means of conditions on the Taylor coefficients of $f$. As an application, we also find conditions such that the Cesáro means of order $\beta$ of $f(z)$ are close-to-convex and starlike in $\bbfD$.
MSC 2000:
*42A32 Trigonometric series of special types

Keywords: positive trigonometric sum; Cesàro means; close-to-convex functions; starlike functions

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