An extension of the Schwarz inequality in inner product spaces

Abstract

\begin{abstract}
We extend the improved Schwarz inequality of Dragomir \cite[Theorem 2]{D17} to any power $p\geq 2$,
\begin{equation*}
\|x\|^p\|y\|^p - |\langle x, y \rangle|^p \geq \left|\det
\begin{bmatrix}
|\langle x, e \rangle|&\quad& (\|x\|^p-|\langle x, e \rangle|^p)^{1/p} \\&\\
|\langle y, e \rangle|&\quad& (\|y\|^p-|\langle y, e \rangle|^p)^{1/p}
\end{bmatrix} \right|^p
\end{equation*}
for any vectors $x$, $y$, $e$ $\in \mathbb{C}^n$ with $\|e\|=1$.
Applications to n-tuples of complex numbers are also included.
\end{abstract}