Problem: For [MATH] {z} \in \mathbb{C}, |{z}|<1,[/MATH] show that [MATH] |\exp(z)| \leq \frac{1}{1-\Re(z) } [/MATH]
My approach: First I noticed that [MATH] |\exp(z)| = \exp(\Re(z)) [/MATH].
Saying that [MATH] a = \Re(z) [/MATH] it's true by definition that [MATH] \exp(a) = \lim_{n \rightarrow \infty} (1 + \frac{a}{n})^{n} [/MATH],
so I tried to use [MATH] \exp(a) \geq (1 + a) [/MATH] and [MATH] a < 1 [/MATH] to arrive at [MATH] \exp(a) \leq (1 - a)^{-1} [/MATH].
I don't know if this is the right approach and I somehow really confused myself ?
My approach: First I noticed that [MATH] |\exp(z)| = \exp(\Re(z)) [/MATH].
Saying that [MATH] a = \Re(z) [/MATH] it's true by definition that [MATH] \exp(a) = \lim_{n \rightarrow \infty} (1 + \frac{a}{n})^{n} [/MATH],
so I tried to use [MATH] \exp(a) \geq (1 + a) [/MATH] and [MATH] a < 1 [/MATH] to arrive at [MATH] \exp(a) \leq (1 - a)^{-1} [/MATH].
I don't know if this is the right approach and I somehow really confused myself ?