Note on the second derivative of bounded analytic functions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909137941561344 |
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| author | Chen, Gangqiang |
| author_facet | Chen, Gangqiang |
| contents | Assume $z_0$ lies in the open unit disk $\mathbb{D}$ and $g$ is an analytic self-map of $\mathbb{D}$. We will determine the region of values of $g''(z_0)$ in terms of $z_0$, $g(z_0)$ and the hyperbolic derivative of $g$ at $z_0$, and give the form of all the extremal functions. In particular, we obtain a smaller sharp upper bound for $|g''(z_0)|$ than Ruscheweyh's inequality for the case of the second derivative. Moreover, we use a different method to obtain Sz{á}sz's inequality, which provides a sharp upper bound for $|g''(z_0)|$ depending only on $|z_0|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10213 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Note on the second derivative of bounded analytic functions Chen, Gangqiang Complex Variables 30C80, 30F45 Assume $z_0$ lies in the open unit disk $\mathbb{D}$ and $g$ is an analytic self-map of $\mathbb{D}$. We will determine the region of values of $g''(z_0)$ in terms of $z_0$, $g(z_0)$ and the hyperbolic derivative of $g$ at $z_0$, and give the form of all the extremal functions. In particular, we obtain a smaller sharp upper bound for $|g''(z_0)|$ than Ruscheweyh's inequality for the case of the second derivative. Moreover, we use a different method to obtain Sz{á}sz's inequality, which provides a sharp upper bound for $|g''(z_0)|$ depending only on $|z_0|$. |
| title | Note on the second derivative of bounded analytic functions |
| topic | Complex Variables 30C80, 30F45 |
| url | https://arxiv.org/abs/2403.10213 |