Estimates for the Complex Exponential and Its Continuity

In the setting of Complex Analysis: Standing Notation, let $\exp:\mathbb{C}\to\mathbb{C}$ be the complex exponential function, and let $z\in\mathbb{C}$.

(i) Remainder bounds. For every $h\in\mathbb{C}$,

$$|\exp h-1|\le|h|\,\exp_{\mathbb{R}}(|h|),\qquad |\exp h-1-h|\le|h|^{2}\,\exp_{\mathbb{R}}(|h|).$$

(ii) Continuity. $\exp$ is a continuous function from the metric space $(\mathbb{C},d_{\mathbb{C}})$ to itself.

(iii) Difference quotients. For every $h\in\mathbb{C}$ with $h\neq0$,

$$\left|\frac{\exp(z+h)-\exp z}{h}-\exp z\right|\le|\exp z|\,|h|\,\exp_{\mathbb{R}}(|h|).$$