On a trigonometric inequality of Vinogradov
Here the author improves a result of \textit{T. Cochrane} [J. Number Theory 27, 9-16 (1987; Zbl 0629.10030)]: Let \(m\) and \(n\) be positive integers with \(m>1\). Set NEWLINE\[NEWLINEf(m,n)= \sum_{s=1}^{m-1} \left(\biggl|\sin \frac{\pi an}{m}\biggr|\biggl/ \sin\frac{\pi a}{m} \right).NEWLINE\]NEWLINE Theorem 1: For any positive integers \(m,n\) with \(m>1\), we have NEWLINE\[NEWLINEf(m,n)< \frac{4m}{\pi^2} \biggl( \log m+ \frac 34+ \gamma- \log\frac\pi 2 \biggr)+ \frac 2\pi \biggl( 2-\frac 1\pi \biggr).NEWLINE\]NEWLINE Theorem 2: For any positive integers \(m>1\), we have NEWLINE\[NEWLINE\frac 1m \sum_{n=1}^m f(m,n)< \frac{4m}{\pi^2} \biggl( \log m+ \gamma- \log \frac\pi 2 \biggr)+ \frac 2\pi \biggl( 2- \frac 1\pi \biggr)- \frac{1}{6m} \biggl(1- \frac 1m \biggr).NEWLINE\]
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