There is some $a$, $b$,$c$, $d$ and $e$, where $a\neq0$, $b\neq 0$ and $c\neq 0$ such that $$t^a\tau^b+r^ct^d\tau^e \leq tr+\tau r^\alpha$$ for all $t>0$, $r>0$, $\tau>0$ and $\alpha \in (0,\frac{1}{2})$? I am trying study the critical points of $f(t,r,\tau)=tr+\tau r^\alpha -t^a \tau^b-r^c$. I really don't know if this holds, but i wold like very much.
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