For some different types of operators on a Hilbert space, we present
new high-power operator inequalities, and their corresponding operator inequalities
involving spectral radii of operators. We prove that each such operator
inequality is equivalent to the Cauchy-Schwarz inequality. In particular,
we show that Halmos’ two operator inequalities, Reid’s inequality, and many
others hold easily. We obtain a new generalized Löwner inequality, and a short
proof of the classical Löwner-Heinz inequality is given.