## Abstract

It is known that the sum of the sine series \(g\left( {{\rm{b,}}\,x} \right) = \sum\nolimits_{k = 1}^\infty {{b_k}} \)*b*_{k} sin *kx* whose coefficients constitute a convex sequence **b** is positive on the interval (0, π). To estimate its values in a neighborhood of zero, Telyakovskii used the piecewise continuous function

He showed that the difference *g***(b**, *x*) − (*b*_{m}(_{x})/2)cot(*x*/2) in a neighborhood of zero admits a two-sided estimate in terms of the function *a*(**b**,*x*) with absolute constants. The exact values of these constants for the class of convex sequences **b** are obtained in this paper.

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## References

- 1.
S. A. Telyakovskiĭ, “On the behavior near the origin of the sine series with convex coefficients,” Publ. Inst. Math. (Beograd) (N. S.)

**58**(72), 43–50 (1995). - 2.
S. A. Telyakovskii, “On the behavior of the sine series near zero,” Makedon. Akad. Nauk. Umet. Oddel. Mat.-Tehn. Nauk. Prilozi

**21**, 47–54 (2002). - 3.
S. Aljančić, R. Bojanić, and M. Tomić, “Sur le comportement asymptotique au voisinage de zéro des séries trigonométriques de sinus à coefficients monotones,” Acad. Serbe Sci. Publ. Inst. Math.

**10**, 101–120 (1956). - 4.
A. Zygmund,

*Trigonometric Series*(Cambridge Univ. Press, Cambridge, 1959, 1960; Mir, Moscow, 1965), Vols. 1, 2. - 5.
E. Seneta,

*Regularly Varying Functions*(Springer-Verlag, Berlin-Heidelberg-New York, 1976; Nauka, Moscow, 1985). - 6.
S. A. Telyakovskii, “On the behavior of sine series with convex coefficients near the origin,” Dokl. Dokl. Akad. Nauk SSSR

**56**(3), 913–914 (1997) [Soviet Math. Dokl.**357**(4), 462–463 (1997)]. - 7.
A. Yu. Popov, “Estimates of the sums of sine series with monotone coefficients of certain classes,” Mat. Zametki

**74**(6), 877–888 (2003) [Math. Notes**74**(6), 829–840 (2003)]. - 8.
A. P. Solodov, “A sharp lower bound for the sum of a sine series with convex coefficients,” Mat. Sb.

**207**(12), 124–158 (2016) [Sb. Math.**207**(12), 1743–1777 (2016)].

## Funding

This work was supported by the Russian Foundation for Basic Research under grant 20-01-00584.

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Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 6, pp. 906–921.

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Solodov, A.P. Exact Constants in Telyakovskii’s Two-Sided Estimate of the Sum of a Sine Series with Convex Sequence of Coefficients.
*Math Notes* **107, **988–1001 (2020). https://doi.org/10.1134/S0001434620050314

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### Keywords

- sine series with monotone coefficients
- convex sequence
- slowly varying sequence