In the theory of transcendental numbers, frequent use is made of a certain inequality which establishes a lower bound for the height of a product of polynomials in terms of the heights of the factors. A particularly general and accurate form of this inequality was proved by A. O. Gelfond. In this note, Mahler gives a new proof for Gelfond's formula and also shows a similar, but simpler, inequality for the length of a product of polynomials. \par Reprint of the author's paper [J. Lond. Math. Soc. 37, 341--344 (1962; Zbl 0105.06301)].