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Abstract
This paper aims to present some inequalities for unitarily invariant norms. In section 2, we give a refinement of the Cauchy-Schwarz inequality for matrices. In section 3, we obtain an improvement for the result of Bhatia and Kittaneh (Linear Algebra Appl. 308 (2000) 203-211). In section 4, we establish an improved Heinz inequality for the Hilbert-Schmidt norm. Finally, we present an inequality involving positive definite matrix and Hilbert-Schmidt norm. Then we use it to discuss the conjecture on the Hilbert-Schmidt norm of matrices proposed by Sloane and Harwit and the conjecture is proved for some special matrices.
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