  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\|\frac{(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 $\textrm{Re}(g)>0$.
