New improvements of some classical inequalities

In this paper, we establish an inequality for scalars, which we then apply to refine some classical inequalities for inner product and numerical raduis. For example, we establish that for any \(\mathcal {E}\in \mathcal {B}(\mathcal {H}),\) $\mathcal{E}\in \mathcal{B}(\mathcal{H}),$ \(u,v\in \mathcal {H},\) $u,v\in \mathcal{H},$ and \(0\le \theta \le 1\) $0\le \theta \le 1$ , \(\begin{aligned} |\langle \mathcal {E} u,v\rangle |^2&\le \mathcal {U}_{(n,\xi )}\left( \eta ,|\langle \mathcal {E}u,v\rangle |,\sqrt{\left\langle |\mathcal {E}|^{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}^*\right| ^{2(1-\theta )} v, v\right\rangle }\right) \\ &\le \left\langle |\mathcal {E}|^{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}^*\right| ^{2(1-\theta )} v, v\right\rangle . \end{aligned}\) $$\begin{array}{cc}\hfill {|⟨\mathcal{E}u,v⟩|}^2& \le {\mathcal{U}}_{(n,\xi )}\left(\eta ,|⟨\mathcal{E}u,v⟩|,,\sqrt{〈{|\mathcal{E}|}^{2\theta },u,,,u〉〈{\left|{\mathcal{E}}^{\ast }\right|}^{2(1-\theta )},v,,,v〉}\right)\hfill \\ \hfill & \le 〈{|\mathcal{E}|}^{2\theta },u,,,u〉〈{\left|{\mathcal{E}}^{\ast }\right|}^{2(1-\theta )},v,,,v〉.\hfill \end{array}$$ Moreover, we have \(\left( \mathcal {U}_{(n,\xi )}\left( \eta ,|\langle \mathcal {E}u,v\rangle |,\sqrt{\left\langle |\mathcal {E}|^{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}^*\right| ^{2(1-\theta )} v, v\right\rangle }\right) \right) _{n \geqslant 0}\) ${\left({\mathcal{U}}_{(n,\xi )},\left(\eta ,|⟨\mathcal{E}u,v⟩|,,\sqrt{〈{|\mathcal{E}|}^{2\theta },u,,,u〉〈{\left|{\mathcal{E}}^{\ast }\right|}^{2(1-\theta )},v,,,v〉}\right)\right)}_{n⩾0}$ is an increasing sequence satisfying \(\begin{aligned} \lim \limits _{n \rightarrow +\infty } \mathcal {U}_{(n,\xi )}\left( \eta ,|\langle \mathcal {E}u,v\rangle |,\sqrt{\left\langle |\mathcal {E}|^{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}^*\right| ^{2(1-\theta )} v, v\right\rangle }\right) = \left\langle |\mathcal {E}|^{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}^*\right| ^{2(1-\theta )} v, v\right\rangle , \end{aligned}\) $$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}{\mathcal{U}}_{(n,\xi )}\left(\eta ,|⟨\mathcal{E}u,v⟩|,,\sqrt{〈{|\mathcal{E}|}^{2\theta },u,,,u〉〈{\left|{\mathcal{E}}^{\ast }\right|}^{2(1-\theta )},v,,,v〉}\right)=〈{|\mathcal{E}|}^{2\theta },u,,,u〉〈{\left|{\mathcal{E}}^{\ast }\right|}^{2(1-\theta )},v,,,v〉,\end{array}$$ which presents a novel refinement of the well-known mixed Schwartz inequality. Our results extend and refine well-established inequalities found in the literature.