If $0\leq a, b, c \leq R$ and $$ 0\leq a+b-2c \leq \epsilon $$ how can we upper bound the quantity $$ a^n + b^n -2c^n $$ where $n\geq1$ is an integer.
My crude bound, assuming $a-c \geq 0$ and $b-c \geq0$, is $\epsilon^n$, but I want to know if there is a nice way to show that $a^n + b^n - 2c^n \leq \epsilon^n$ without any assumption (or maybe other relevant results)