A refinement of the Cauchy-Schwarz inequality accompanied by new numerical radius upper bounds

Abstract

This present work aims to ameliorate the celebrated Cauchy-Schwarz inequality and provide several new consequences associated with the numerical radius upper bounds of Hilbert space operators. More precisely, for arbitrary $a,b\in H$ and $\alpha \geq 0$, we show that
\vspace*{-.06in}
\begin{equation*}
\begin{array}{l@{}>{\displaystyle}l}
{\left|
{\left\langle {a,b} \right\rangle } \right|^2} \,\,&\,\,\le
\frac{1}{\alpha+1
}\left\| a \right\|\left\| b \right\|\left| {\left\langle {a,b}
\right\rangle } \right| + \frac{\alpha}{\alpha+1 }{\left\| a
\right\|^2}{\left\| b \right\|^2}\\
\vspace*{-.1in}\\
\,\,&\,\,\le {\left\|a\right\|}^{2}{\left\|b\right\|}^{2}.
\end{array}
\end{equation*}
As a consequence, we provide several new upper bounds for the numerical radius that refine and generalize some of Kittaneh's results in [A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix. Studia Math. 2003;158:11–17] and [Cauchy-Schwarz type inequalities and applications to numerical radius inequalities. Math. Inequal. Appl. 2020;23:1117–1125], respectively. In particular, for arbitrary $A,B\in B(H)$ and $\alpha \geq 0$, we show the following sharp upper bound
\vspace*{-.06in}
\begin{equation*}
{w^2}\left( {{B^*}A} \right) \le
\frac{1}{{2\alpha + 2}}\left\| {{{\left| A \right|}^2} + {{\left| B
\right|}^2}} \right\|w\left( {{B^*}A} \right) + \frac{\alpha
}{{2\alpha + 2}}\left\| {{{\left| A \right|}^4} + {{\left| B
\right|}^4}} \right\|,
\end{equation*}
with equality holds when $A=B=\scriptscriptstyle{\begin{pmatrix}
0 & 1 \\
0 & 0 \\
\end{pmatrix}}$.
It is also worth mentioning here that some specific values of $\alpha \geq 0$ provide more accurate estimates for the numerical radius.
Finally, some related upper bounds are also provided.