Monotonic trigonometric sums and coefficients of Bloch functions (Q1282985)

We prove that for every positive integer \(n\), the following inequality holds true \[ \frac{d}{d\theta} \Biggl\{ \frac{\sin^4 \frac\theta 2}{\sin\theta} \sum_{k=1}^n \frac{\sin k\theta}{k^\alpha} \Biggr\}>0, \quad 0<\theta<\pi, \] when \(\alpha\geq 3\). This inequality is false for appropriate \(n\) and \(\theta\) when \(\alpha<3\). Through a theorem of Andreev and Duren this result gives new information about the coefficients of Bloch functions.