Define a transformation of the prime counting function, $\pi(k)$, by $$J: (0,1) \longrightarrow (0,1) $$

where

$$J(x)= \lim_{r \to \infty} \frac{\int_2^{rx} \pi(k) \, dk }{\int_2^r \pi(k) \, dk}$$

I'm interested in optimizing the following over $s,t \in \Bbb R_+$

$$ x^s\le J(x) \le x^t $$

Let $t=1.$

Does the inequality hold for $s=19/10$?