Polar decomposition approach to Reid's inequality

If and is Hermitian, then holds for all , which is known as Reid's inequality and was sharpened by Halmos in which was replaced by , the spectral radius of . In this article we present generalizations of Reid's and Halmos' inequalities via polar decomposition approach. Conditions on and are relaxed. Theorem regards Reid-type inequalities, and Theorem 2 contains Halmos-type inequalities.