We prove the Löwner-Heinz inequality, via the Cordes inequality, for elements a,b>0 of a unital hermitian Banach *-algebra A. Letting p be a real number in the interval (0,1], the former asserts that a^p \le b^p if a \le b, a^p < b^p ifa<b , provided that the involution of A is continuous, and the latter that s(a^pb^p) \le s(ab)^p, where s(x)=r(x^*x)^{1/2} andr(x) is the spectral radius of an element x.