For positive integer $n$ and positive reals $x,y,a,b$
$x a^n + y b^n < (x+\frac{a}{3})\int (a+x+\frac{1+a}{b})^n da + (y+\frac{b}{3})\int (b+y+\frac{1+b}{a})^ndb+\frac{9}{4} $
Is this true ?
What is the easiest way to decide this ?
In inequalities is there a priority in proofs for calculus tools vs algebraic methods ?
I sometimes find inequalities puzzling. Some advice is welcome.