  Several unitarily invariant norm inequalities and numerical radius
inequalities for Hilbert space operators are studied. We investigate some
necessary and sufficient conditions for the parallelism of two bounded
operators. For a finite rank operator $A,$ it is shown that \begin{eqnarray*}
\|A\|_{p} &\leq &\left(\textit{rank} \, A\right)^{1/{2p}} \|A\|_{2p} \,\, \leq
\,\, \left(\textit{rank} \, A\right)^{{(2p-1)}/{2p^2}} \|A\|_{2p^2}, \quad
\textit{for all $p\geq 1 $} \end{eqnarray*} where $\|\cdot\|_p$ is the Schatten
$p$-norm. If $\{ \lambda_n(A) \}$ is a listing of all non-zero eigenvalues
(with multiplicity) of a compact operator $A$, then we show that
\begin{eqnarray*} \sum_{n} \left|\lambda_n(A)\right|^{p} &\leq& \frac12 \|
A\|_{ p}^{ p} + \frac12 \| A^2\|_{p/2}^{p/2}, \quad \textit{for all $p\geq 2$}
\end{eqnarray*} which improves the classical Weyl's inequality $\sum_{n}
\left|\lambda_n(A)\right|^{p} \leq \| A\|_{ p}^{ p}$ [Proc. Nat. Acad. Sci. USA
1949]. For an $n\times n$ matrix $A$, we show that the function $p\to
n^{-{1}/{p}}\|A\|_p$ is monotone increasing on $p\geq 1,$ complementing the
well known decreasing nature of $p\to \|A\|_p.$ \indent As an application of
these inequalities, we provide an upper bound for the sum of the absolute
values of the zeros of a complex polynomial. As another application we provide
a refined upper bound for the energy of a graph $G$, namely, $\mathcal{E}(G)
\leq \sqrt{2m\left(\textit{rank Adj(G)} \right)},$ where $m$ is the number of
edges, improving on a bound by McClelland in $1971$.
