Geodesic
The sharp constant in the estimate of the Rogozinski sums deviation in terms of the second modulus of continuity
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 25, Tome 247 (1997), pp. 26-45

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The sharp constant (uniformly in $n$) is found in a Jackson-type inequality involving the Rogozinski sums of order $n$ and the second modulus of continuity with the step $\pi/(n+1)$. 
@article{ZNSL_1997_247_a2,
     author = {O. L. Vinogradov},
     title = {The sharp constant in the estimate of the {Rogozinski} sums deviation in terms of the second modulus of continuity},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {26--45},
     publisher = {mathdoc},
     volume = {247},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_247_a2/}
}
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O. L. Vinogradov. The sharp constant in the estimate of the Rogozinski sums deviation in terms of the second modulus of continuity. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 25, Tome 247 (1997), pp. 26-45. http://geodesic.mathdoc.fr/item/ZNSL_1997_247_a2/