An alternate proof of Gerretsen's inequalities
The author proves the \(16Rr-5r^2\leq s^2\) part of Gerretsen's inequality by using Schur's inequality and the \(s^2\leq 4R^2+4Rr+3r^2\) part by using an inequality that holds for any three real numbers together with an identity involving \(\cos A \cos B \cos C\). He then considers equivalent forms and shows that Ono's inequality for acute triangles (\(27(-a^2+b^2+c^2)^2(a^2-b^2+c^2)^2(a^2+b^2-c^2)^2\leq (4A)^6\)), Blundon's inequality (\(s\leq 2R+ (3\sqrt{3}-4)r\)), and the Hadwiger-Finsler inequality can be deduced from Gerretsen's inequality.
scientific article; zbMATH DE number 6694041