[SOME PROPERTIES OF THE SERIES OF COMPOSED NUMBERS, p2](http://www.emis.de/journals/EM/expmath/volumes/11/11.2/Garunkstis301.pdf) 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](http://files.ele-math.com/preprints/mia-15-52-pre.pdf) 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.



> 
> 1. Is this reasoning correct?
> 2. Can (2) be proved?
> 
> 
> 


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



Can these logarithmic inequalities show existence of a prime between (x-1)^2 and x^2
Note that, in the second paper, the inequality $$x\pi(x)<2\pi(x^2)$$
is derived from the inequality
$$xy\pi(x)\pi(y)>4\pi^2(xy)$$
by setting $x=y$. The proof for theorem 3 seems correct to me, so the sign flip is definitely a typo :)


