Assume $n>1$ is natural and set $$f_n(a,b):=\operatorname{Re}((2a+ib)^{2n+1}).$$ Prove that for every coprime pair $2a,b\in\mathbb{Z}$: $|f_n(a,b)|>b$.
Note that we have $b|f_n(a,b)$ so the only thing required is to show the inequality is strong. Noting that $|f_n(a,b)|$ behaves exponential (from checking specific values) in $n$ it seems trivial, yet a proof eludes me.
We can expand it: $$f_n(a,b)=b\sum_{k=0}^n{{2n+1}\choose{2k}}(-4)^ka^{2k}b^{2(n-k)}$$ But nothing from here.
Induction seems impossible due to frequent sign changes; Stirling's bound gives no help either.
Thanks in advance