Solution for a particular inequality

Question

I'm following up on this question on MathOverflow. By taking some particular functions, $a(.),\,b(.)$ and $K(.,.)$.

This problem is equivalent to the following inequality:

$$4\left \| r \right \|_{\infty}\big(t^\frac{5}{6}-t^\frac{1}{3}ln(1+t^\frac{1}{2})\big)\leq r(t),\,t\in[0,1].\:(I)$$

I'm looking for a function $r \in \mathcal{C}\big(I,(0, \infty)\big)$ which verifies $(I)$.

Any help is appreciated.

Answer 1

This is not possible. Define $$ f(t) = 4 \, (t^{5/6} - t^{1/3} \ln(1 + t^{1/2})).$$ Then, $f$ is continuous and $f(1) = 4 (1 - \ln(2)) \approx 1.227$. Hence, your inequality implies $$ 1.1 \, \|r\|_\infty \le r(t)$$ for all $t$ in the neighborhood of $1$, but this is not possible.