A converse to some eigenvalue inequalities involving convex functions of matrices

In this paper, we present a converse for some eigenvalue inequalities involving convex functions. In particular, we show that if $Cinmathbb{M}_n$ is an isometry, then$$lambda_k^downarrowleft( left(C^*XCight)^qight) geq lambda_k^downarrowleft(C^*X^qC + (lambda_1(X)- lambda_n(X)^q)ight),quad (k=1,dots,n)$$ for every $qin[1,2]$ and every positive matrix $X$.