In this paper we prove among others that $\begin{array}{rl}∥f\left(y\right)z-zf\left(x\right)∥& \le \frac{1}{2\pi }∥yz-zx∥{\int }_{\gamma }\left|f\left(\xi \right)\right|∥R_y\left(\xi \right)∥∥R_x\left(\xi \right)∥\left|\mathrm{d}\xi \right|\\ & \le \frac{1}{2\pi }∥yz-zx∥{\int }_{\gamma }\frac{\left|f\left(\xi \right)\right|\left|\mathrm{d}\xi \right|}{\left(\left|\xi \right|-∥y∥\right)\left(\left|\xi \right|-∥x∥\right)},\end{array}$ where $f:D\subset \mathbb{C}\to \mathbb{C}$ is an analytic function on the domain D, x, y, $z\in \mathcal{B}$ with $\sigma \left(x\right),$ $\sigma \left(y\right)\subset D,$ $R_y\left(\cdot \right),$ $R_x\left(\cdot \right)$ are the resolvent functions for the elements y and x, and γ is a closed rectifiable path in D and such that $\sigma \left(x\right),$ $\sigma \left(y\right)\subset \mathrm{i}\mathrm{n}\mathrm{s}\left(\gamma \right).$ Applications for the exponential function on the Banach algebra $\mathcal{B}$ are also given.

中文翻译：

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Banach 代数中解析函数的一些不等式

在本文中，我们证明了$\begin{array}{rl}∥F\left(\mathrm{是的}\right)z-zF\left(X\right)∥& \le \frac{1}{2\pi }∥\mathrm{是的}z-zX∥{\int }_{\gamma }\left|F\left(\xi \right)\right|∥R_{\mathrm{是的}}\left(\xi \right)∥∥R_X\left(\xi \right)∥\left|\mathrm{d}\xi \right|\\ & \le \frac{1}{2\pi }∥\mathrm{是的}z-zX∥{\int }_{\gamma }\frac{\left|F\left(\xi \right)\right|\left|\mathrm{d}\xi \right|}{\left(\left|\xi \right|-∥\mathrm{是的}∥\right)\left(\left|\xi \right|-∥X∥\right)},\end{array}$在哪里$F：D\subset \mathbb{C}\to \mathbb{C}$是域D , x , y上的解析函数，$z\in \mathcal{乙}$和$\sigma \left(X\right),$ $\sigma \left(\mathrm{是的}\right)\subset D,$ $R_{\mathrm{是的}}\left(\cdot \right),$ $R_X\left(\cdot \right)$是元素y和x的分解函数，并且γ是D中的闭合可校正路径，因此$\sigma \left(X\right),$ $\sigma \left(\mathrm{是的}\right)\subset \mathrm{一世}\mathrm{n}\mathrm{s}\left(\gamma \right).$指数函数在 Banach 代数上的应用$\mathcal{乙}$也给出了。