MathDB

Let

a,b\in \mathbb{R}

and

z\in \mathbb{C}\backslash \mathbb{R}

so that

\left| a-b \right|=\left| a+b-2z \right|

. a) Prove that the equation

{{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}={{\left| a-b \right|}^{x}}

, with the unknown number

x\in \mathbb{R}

, has a unique solution. b) Solve the following inequation

{{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}\le {{\left| a-b \right|}^{x}}

, with the unknown number

x\in \mathbb{R}

. The Mathematical Gazette

Problem Statement