Problem

Source: 2016 China South East Mathematical Olympiad Grade 11 Problem 5

Tags: complex numbers, inequalities, algebra, natural logarithm, China Southeast MO



Let a constant $\alpha$ as $0<\alpha<1$, prove that: $(1)$ There exist a constant $C(\alpha)$ which is only depend on $\alpha$ such that for every $x\ge 0$, $\ln(1+x)\le C(\alpha)x^\alpha$. $(2)$ For every two complex numbers $z_1,z_2$, $|\ln|\frac{z_1}{z_2}||\le C(\alpha)\left(|\frac{z_1-z_2}{z_2}|^\alpha+|\frac{z_2-z_1}{z_1}|^\alpha\right)$.