SOME PROPERTIES OF THE SERIES OF COMPOSED NUMBERS, p2 gives the bounds

$$l(x)=\frac{x}{\log(x)-28/29}<\pi(x) < u(x)=\frac{x}{\log(x)-1.12} \qquad (1)$$

for $x \geq 3299$.

TWO GENERALIZATIONS OF LANDAU’S INEQUALITY, p6 gives

$$x\pi(x) < 2 \pi(x^2) $$ for $x \geq 67$.

To show $\pi(x^2) > x\pi(x)/2 > \pi((x-1)^2)$ tried the bounds in (1) leading to:

$$xl(x) /2 > u((x-1)^2) \iff $$ $$ \frac{29\, x^{2}}{2 \, {\left(29 \, \log\left(x\right) - 28\right)}} - \frac{{\left(x - 1\right)}^{2}}{\log\left({\left(x - 1\right)}^{2}\right) - 1.12} >0 \qquad (2)$$

Experimentally the LHS of (2) is increasing up to $10^{120}$ and is quite large.

- Is this reasoning correct?
- Can (2) be proved?

All CAS I tried failed to solve (2) though the limit at infinity is $+\infty$.