From where you left off, we want to prove this:
(\frac{a+b}{2})^{k+1}\leq (\frac{a^{k+1}+b^{k+1}}{4})+(\frac{ab^k + ba^k}{4}) \leq (\frac{a^{k+1}+b^{k+1}}{2})
The above would be true if and only if:
(\frac{ab^k + ba^k}{4})\leq (\frac{a^{k+1}+b^{k+1}}{4}) \\ \Leftrightarrow ab^k + ba^k \leq...