Enhancement of the Cauchy-Schwarz Inequality and Its Implications for Numerical Radius Inequalities
| Autor: | Nayak, Raj Kumar |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | In this article, we establish an improvement of the Cauchy-Schwarz inequality. Let $x, y \in \mathcal{H},$ and let $f: (0,1) \rightarrow \mathbb{R}^+$ be a well-defined function, where $\mathbb{R}^+$ denote the set of all positive real numbers. Then \[|\langle x, y \rangle|^2 \leq \frac{f(t)}{1+f(t)} \|x\|^2 \|y\|^2 + \frac{1}{1+ f(t)} |\langle x, y \rangle | \|x\|\|y\|. \] We have applied this result to derive new and improved upper bounds for the numerical radius. |
| Databáze: | arXiv |
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