If Second Hardy-Littlewood Conjecture is true then we can claim that $\pi(x)-\pi(y)\leq \pi(x-y)$. Thus the conjecture gives an upper bound for the number of primes between $x$ and $y$. I have found some similar results that gives an upper bound for the number of primes in some of the papers (for example see this paper).

But my question is:

is there any conjecture, say of the form $\ \pi(x)-\pi(y)\geq a\pi(f(x,y)),\ $ which holds for all sufficiently large $x$ and $y,$ and where $\ a\ $ is a constant greater than $1$?

For a more clarification of the conjecture and consequences let us just observe that if $a=1$, $f(x,y)=\sqrt{x}-\sqrt{y}$ then we can have $\pi(x)-\pi(y)\geq \pi \left(\sqrt{x}-\sqrt{y}\right)$. Now putting $x=(n+1)^2$ and $y=n^2$ we see that a consequence of the inequality is Legendre Conjecture.

Basically, most of the conjectures regarding the distribution of primes between two numbers asks us to prove that there is at least one prime between the two said numbers. In this aspect I think that this type of inequality may be particularly helpful.

In a similar manner some other conjectures regarding the distribution of prime numbers between may be tackled very effectively if a good expression for $f$ be found out.

If so, then please give a link of the source in your answer (or comment).