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constant $\alpha$

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

August 6, 2016
complex numbersinequalitiesalgebranatural logarithmChina Southeast MO

Problem Statement

Let a constant α\alpha as 0<α<10<\alpha<1, prove that: (1)(1) There exist a constant C(α)C(\alpha) which is only depend on α\alpha such that for every x≥0x\ge 0, ln⁡(1+x)≤C(α)xα\ln(1+x)\le C(\alpha)x^\alpha. (2)(2) For every two complex numbers z1,z2z_1,z_2, ∣ln⁡∣z1z2∣∣≤C(α)(∣z1−z2z2∣α+∣z2−z1z1∣α)|\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).
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