On a trigonometric inequality of vinogradov

The sum f(m, n)=∑a=1m−1(|sinπanm||sinπam|) arises in bounding incomplete exponential sums. In this article we show that for positive integers m, n with m>1, f(m, n)<(4π2) m log m+0.38m+0.608+0.116 d2m, where d=(m, n). This improves earlier bounds for f(m, n). The constant 4π2 in the main term is shown to be best possible.