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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2405.19698 |
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| _version_ | 1866911894184394752 |
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| author | Nayak, Raj Kumar |
| author_facet | Nayak, Raj Kumar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19698 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enhancement of the Cauchy-Schwarz Inequality and Its Implications for Numerical Radius Inequalities Nayak, Raj Kumar Functional Analysis 47A12, 47A30 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. |
| title | Enhancement of the Cauchy-Schwarz Inequality and Its Implications for Numerical Radius Inequalities |
| topic | Functional Analysis 47A12, 47A30 |
| url | https://arxiv.org/abs/2405.19698 |