Inequalities for convex cyclic polygons
Let A be the area of a convex cyclic n-gon, R its circumradius and s its semiperimeter. The main result of the paper is the following theorem: Suppose that, for \(x>0\), f(x) is positive and non-decreasing and f(x)/x is non-increasing. If \(a_ 1,a_ 2,...,a_ n\) are the sides of a convex cyclic polygon, then \(f(a_ 1),f(a_ 2),...,f(a_ n)\) are the sides of another convex cyclic polygon, for which \(A/s^ 2\), s/R, and \(A/R^ 2\) are at least as large as the corresponding ratios for the first polygon. Strict inequality holds for all three ratios, unless f(x) is proportional to x at \(a_ 1,a_ 2,...,a_ n\) or \(a_ 1=a_ 2=...=a_ n\) holds.
scientific article; zbMATH DE number 4173789