Exact Constants in Telyakovskii's Two-Sided Estimate of the Sum of a Sine Series with Convex Sequence of Coefficients

It is known that the sum of the sine series $g(\mathbf b,x)=\sum_{k=1}^\infty b_k\sin kx$ whose coefficients constitute a convex sequence $\mathbf b$ is positive on the interval $(0,\pi)$. To estimate its values in a neighborhood of zero, Telyakovskii used the piecewise continuous function
$$ \sigma(\mathbf b,x)=\frac1{m(x)}\sum_{k=1}^{m(x)-1}k^2(b_k-b_{k+1}),\qquad m(x)=\biggl[\frac\pi x\biggr]. $$
He showed that the difference $g(\mathbf b,x)-(b_{m(x)}/2)\operatorname{cot}(x/2)$ in a neighborhood of zero admits a two-sided estimate in terms of the function $\sigma(\mathbf b,x)$ with absolute constants. The exact values of these constants for the class of convex sequences $\mathbf b$ are obtained in this paper.