Some inequalities for unitarily invariant norm

We shall prove the inequalities|||(A+B)(A+B)∗|||⩽|||AA∗+BB∗+2AB∗|||⩽|||(A-B)(A-B)∗+4AB∗|||for all n×nn×n complex matrices A,BA,B and all unitarily invariant norms ∣∣∣·∣∣∣.∣∣∣·∣∣∣. If further A,BA,B are positive definite it is proved that∏j=1kλj(A♯αB)⩽∏j=1kλj(A1-αBα),1⩽k⩽n,0⩽α⩽1,where ♯α♯α denotes the operator means considered by Kubo and Ando and λj(X),λj(X),1⩽j⩽n,1⩽j⩽n, denote the eigenvalues of XX arranged in the decreasing order whenever these all are real. A number of inequalities are obtained as applications.