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 E is an element of B(H),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}\in \mathcal {B}(\mathcal {H}),$$\end{document}u,v is an element of H,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u,v\in \mathcal {H},$$\end{document} and 0 <=theta <= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le \theta \le 1$$\end{document}, |< Eu,v >|2 <= U(n,xi)eta,|< Eu,v >|,|E|2 theta u,uE & lowast;2(1-theta)v,v <=|E|2 theta u,uE & lowast;2(1-theta)v,v.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\langle \mathcal {E} u,v\rangle |<^>2&\le \mathcal {U}_{(n,\xi )}\left( \eta ,|\langle \mathcal {E}u,v\rangle |,\sqrt{\left\langle |\mathcal {E}|<^>{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}<^>*\right| <^>{2(1-\theta )} v, v\right\rangle }\right) \\ &\le \left\langle |\mathcal {E}|<^>{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}<^>*\right| <^>{2(1-\theta )} v, v\right\rangle . \end{aligned}$$\end{document}Moreover, we have U(n,xi)eta,|< Eu,v >|,|E|2 theta u,uE & lowast;2(1-theta)v,vn >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \mathcal {U}_{(n,\xi )}\left( \eta ,|\langle \mathcal {E}u,v\rangle |,\sqrt{\left\langle |\mathcal {E}|<^>{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}<^>*\right| <^>{2(1-\theta )} v, v\right\rangle }\right) \right) _{n \geqslant 0}$$\end{document} is an increasing sequence satisfying limn ->+infinity U(n,xi)eta,|< Eu,v >|,|E|2 theta u,uE & lowast;2(1-theta)v,v=|E|2 theta u,uE & lowast;2(1-theta)v,v,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim \limits _{n \rightarrow +\infty } \mathcal {U}_{(n,\xi )}\left( \eta ,|\langle \mathcal {E}u,v\rangle |,\sqrt{\left\langle |\mathcal {E}|<^>{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}<^>*\right| <^>{2(1-\theta )} v, v\right\rangle }\right) = \left\langle |\mathcal {E}|<^>{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}<^>*\right| <^>{2(1-\theta )} v, v\right\rangle , \end{aligned}$$\end{document}which presents a novel refinement of the well-known mixed Schwartz inequality. Our results extend and refine well-established inequalities found in the literature.
