Problem 1755 (difficulty: 3/10)

Prove using the behavior of the exponent at the points \(\displaystyle 0\), \(\displaystyle \infty\) and \(\displaystyle 1\) that

\(\displaystyle \left|e^{\frac{z+1}{z-1}}\right|<1 \quad (|z|<1). \)

Give me another random problem!