An extension of the Schwarz inequality in inner product spaces

Yun Ye


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


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