Suppose \(m_1,\ldots, m_n\) are distinct positive integers. Put \(f(x)=-\cos m_1x - \ldots -\cos m_nx\) for real \(x\). Since \(f\) has mean-value zero, its maximum value is a positive number \(K\). Let \(T\) be the number of solutions of the equation \(m_i+m_j=m_k\) in integers \(i, j, k\) between \(1\) and \(n\) inclusive. Using elementary calculus and Schwarz's inequality, the author proves that \[ T=-\frac{2}{3\pi}\int_0^{2\pi}f^2(x)\,dx \geq \frac{n(n-2K^x)}{3K} \] Thus \[ K\geq(\theta+(\theta^2+\tfrac12 n)^{1/2})1 {-1}, \] where \(\theta=3T/2n^{2}\). The author proves an inequality for \[ \frac{4}{\pi}\int_0^{2\pi}r(x)\,dx=3U+4V, \] where \(U\) is the number of solutions of the equation \(m_i+m_j=m_k+m_l\) and \(V\) in integers \(i,j,k,l\) between \(1\) and \(n\) inclusive. Reviewer's remark: In the paper itself, theorems 2, 3 and 4 are stated incorrectly. [For part XIX, see ibid. 32, 90--92 (1956; Zbl 0072.28702).]