Inequalities for sine sums with more variables
The Victoris results in its modified form states that if the real numbers \( a_{1},\dots,a_{n}\) satisfy \[ a_{1}\geq \frac{a_{2}}{2}\geq \dots\geq \frac{a_{n}}{n}>0\text{ and } a_{2k-1}\geq a_{2k} \qquad (0\leq k\leq \frac{n}{k}), \quad (\ast ) \] for \(x_{1},\dots,x_{m}>0\) \ with \ \(x_{1}+\dots +x_{m}<\pi \) then, \[ \ \sum_{k=1}^{n}a_{k}\frac{\sin (kx_{1})\dots\sin (kx_{m})}{k^{m}}>0. \quad(\ast \ast ) \] In this paper, the authors study the above results and prove that, by imposing some weaker conditions on \((\ast \ast )\), strict inequalities \(>\) can be replaced with \(\geq \) and the results will still hold true. Furthermore, \ if condition \((\ast )\) is replaced by \[ \sum_{k=1}^{N}a_{k}\frac{\sin (kt)}{k}>0\qquad(N=1,\dots,n, \; 0<t<\pi ), \] then \((\ast \ast )\) holds true even for a larger region \(x_{1},\dots,x_{m}\in (0,\pi ).\) Many examples are given to demonstrate the usefulness of the results obtained.
scientific article; zbMATH DE number 7462103
scientific article