We develop a new refinement of the Kato’s inequality and using this refinement we obtain several upper bounds for the numerical radius of a bounded linear operator as well as the product of operators, which improve the well known existing bounds. Further, we obtain a necessary and sufficient condition for the positivity of \(2\times 2\) certain block matrices and using this condition we deduce an upper bound for the numerical radius involving a contraction operator. Furthermore, we study the Schatten p-norm inequalities for the sum of two \(n\times n\) complex matrices via singular values, and from the inequalities we obtain the p-numerical radius and the classical numerical radius bounds. We show that for every \(p>0\) , the p-numerical radius \(w_p(\cdot ): \mathcal {M}_n({\mathbb {C}})\rightarrow {\mathbb {R}}\) satisfies \( w_p(T) \le \frac{1}{2} \sqrt{\left\| |T|^{2(1-t)}+|T^*|^{2(1-t)} \right\| \, \big \Vert |T|^{2t}+|T^*|^{2t} \big \Vert _{p/2} } \) for all \(t\in [0,1]\) . Considering \(p\rightarrow \infty \) , we get a nice refinement of the well known classical numerical radius bound \(w(T) \le \sqrt{\frac{1}{2} \left\| T^*T+TT^* \right\| }.\) As an application of the Schatten p-norm inequalities we develop a bound for the energy of a graph. We show that \( \mathcal {E}(G) \ge \frac{2\,m}{ \sqrt{ \max _{1\le i \le n} \left\{ \sum _{j, v_i \sim v_j}d_j\right\} } },\) where \(\mathcal {E}(G)\) is the energy of a simple graph G with m edges and n vertices \(v_1,v_2,\ldots ,v_n\) such that degree of \(v_i\) is \(d_i\) for each \(i=1,2,\ldots ,n.\)
