I posed a conjecture as follows that is a special case of [Second Hardy–Littlewood conjecture](https://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture):



> 
> 
> > 
> > For $n, x \ge 2$ be two integers then:
> >  $$P\_{2n} \ge 2P\_n$$ 
> >  and 
> >  $$\pi(2x) \le 2\pi(x)$$
> > 
> > 
> > 
> 
> 
> 


Where $P\_n$ is $n$-$th$ prime and $\pi(x)$ is [Prime-counting function](https://en.wikipedia.org/wiki/Prime-counting_function)


**My question:** Is the conjecture above contradictory (inconsistent) to the [Prime k-tuple conjecture](https://en.wikipedia.org/wiki/Prime_k-tuple)? I am looking for a proof, comment or reference.



Is $P_{2n} \ge 2P_n$ and $\pi(2x) \le 2\pi(x)$ inconsistent to the Prime k-tuple conjecture?
We have $\pi(2x)<2\pi(x)$ for any $x\geq 11$. For a proof see [here](https://math.stackexchange.com/questions/1738347/proving-that-pi2x-2-pix). Similarly, it follows from standard bounds that $P\_{2n}>2P\_n$ when $n$ is sufficiently large.


