Question 58
Let $a, b \in \mathbb{R}$ and $z \in \mathbb{C} \setminus \mathbb{R}$ such that $|a - b| = |a + b - 2z|$.
a. Prove that there exists an unique real number $x$ which satisfies $|z - a|^x + |\bar{z} - b|^x = |a - b|^x$.
b. Find all real numbers $x$ such that $|z - a|^x + |\bar{z} - b|^x \le |a - b|^x$.
a. Prove that there exists an unique real number $x$ which satisfies $|z - a|^x + |\bar{z} - b|^x = |a - b|^x$.
b. Find all real numbers $x$ such that $|z - a|^x + |\bar{z} - b|^x \le |a - b|^x$.