On Kakeya's Geometric Proof of Enestr\"om-Kakeya's Theorem

Abstract

This paper is devoted to demonstrate Kakeya's geometric proof of his theorem (1912), independently established earlier by Enestr\"om (1893). By calculating centers and radii of the interlacing circles of Kakeya's method, we prove Kakeya's geometric structure, which has not been previously established. We give an equivalent proof, which is based on the construction of internally interlacing circles, which has been geometrically considered by Tomic (1948).