Bounds for the maximum modulus of all positive (or all complex) roots of a polynomial are a fundamental building block of algorithms involving algebraic equations. We apply known results to show which are the salient features of the Lagrange (real) root-bound as well as the related bound by Fujiwara. For a polynomial of degree n, we construct a bound of relative overestimation at most 1.72n which overestimates the Cauchy root by a factor of two at most. This can be carried over to the bounds by Kioustelidis and Hong. Giving a very short variant of a recent proof presented by Collins, we sketch a way to further definite, measurable improvement.
