Summary: The purpose of the present account is to sharpen Heinz's inequality, and to investigate the equality and the bound of the inequality. As a consequence of this we present a Bernstein type inequality for nonselfadjoint operators. The Heinz inequality can be naturally extended to a more general case, and from which we obtain in particular Bessel's equality and inequality. Finally, Bernstein's inequality is extended to \(n\) eigenvectors, and shows that the bound of the inequality is preserved.