Communications of the
Korean Mathematical Society CKMS

In this paper, we present some upper bounds for unitarily invariant norms inequalities. Among other inequalities, we show some upper bounds for the Hilbert-Schmidt norm. In particular, we prove \begin{align*} \|f(A)Xg(B)\pm g(B)Xf(A)\|_2\leq \left\|\tfrac{(I+|A|)X(I+|B|)+(I+|B|)X(I+|A|)}{d_Ad_B}\right\|_2, \end{align*} where $A, B, X\in\mathbb{M}_n$ such that $A$, $B$ are Hermitian with $\sigma (A)\cup\sigma(B)\subset\mathbb{D}$ and $f, g$ are analytic on the complex unit disk $\mathbb{{D}}$, $g(0)=f(0)=1$, $\textrm{Re}(f)>0$ and $\textrm{Re}(g)>0$.

Keywords: $G_1$ operator, unitarily invariant norm, commutator operator, the Hilbert-Schmidt, analytic function

MSC numbers: Primary 15A60; Secondary 30E20, 47A30, 47B10, 47B15