Let \(f(x)=a_ 0+\sum^{\infty}_{n=1}a_ n\quad \cos \quad nx\) and \(\tilde f(x)=\sum^{\infty}_{n=1}a_ n\quad \sin \quad nx,\) where \(a_ 1\neq 0\) and \(a_ n\downarrow 0\) (n\(\to \infty)\) and let \(s_ n(f;x)\), \(s_ n(\tilde f;x)\) denote respective partial sums of these series. The author proves that if \(s_ n(f;x)\geq 0\) (n\(\geq 1)\) and \(x\in (0,\pi)\) then \(a_ n\leq C(f;\beta)n^{-\beta}\) for \(n\geq 1\) and \(0<\beta <1-3\pi /(2+3\pi)\) where the constant C(f;\(\beta)\) depends upon only on the function f and \(\beta\). The results under the condition \(s_ n(\tilde f;x)\geq 0\) have also been obtained. He further shows that the inequality \(a_ n\leq C(\tilde f)/n\) is best possible if \(a_ n- 2a_{n+1}+a_{n+2}\geq 0\) for \(n\geq 1\). The author also proves that one can find a double trigenometric series (*) \(\sum^{\infty}_{n,k=1}a_{nk} \sin nx \sin ky\) such that \(a_{nk}\to 0\) as \(n+k\to \infty\), \(a_{nk}\geq a_{n+1,k}\), \(a_{nk}\geq a_{n,k+1}\) for \(n,k=1,2,...\) and every subsequence \((s_{m_ kn_ k}(x,y))\), where \(m_ k\uparrow \infty\) and \(n_ k\uparrow \infty\), as \(k\to \infty\), of the rectangular partial sums of (*) diverges almost everywhere on \((0,\pi)^ 2\).