A sharp lower bound for the sum of a sine series with convex coefficients

The sum of a sine series $g(\mathbf b,x)=\sum_{k=1}^\infty b_k\sin kx$ with coefficients forming a convex sequence $\mathbf b$ is known to be positive on the interval $(0,\pi)$. Its values near zero are conventionally evaluated using the Salem function $v(\mathbf b,x)=x\sum_{k=1}^{m(x)} kb_k$, $m(x)=[\pi/x]$. In this paper it is proved that $2\pi^{-2}v(\mathbf b,x)$ is not a minorant for $g(\mathbf b,x)$. The modified Salem function $v_0(\mathbf b,x)=x\bigl(\sum_{k=1}^{m(x)-1} kb_k+(1/2)m(x)b_{m(x)}\bigr)$ is shown to satisfy the lower bound $g(\mathbf b,x)>2\pi^{-2}v_0(\mathbf b,x)$ in some right neighbourhood of zero. This estimate is shown to be sharp on the class of convex sequences $\mathbf b$. Moreover, the upper bound for $g(\mathbf b,x)$ is refined on the class of monotone sequences $\mathbf b$.
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