I am looking to prove $$ \frac{ab}{a⁵+b⁵+ab} + \frac{bc}{b⁵+c⁵+bc} + \frac{ac}{a⁵+c⁵+ac} \le1 $$ Where $ a,b,c $ are positive reals with $ abc = 1 $
My initial thought was to apply AM-GM to get $$ a⁵ + b⁵ + ab \ge3(a²b²) $$ and substituting this in because this will give me the maximum value of LHS. But this is giving wrong results.
I am unable to see where I am making mistake, can anyone point out why my thought was wrong?
