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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).

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    $\begingroup$ Sure. Bertrand's postulate is the case $f(x,y) = 2$ if $x > 2y$ and zero otherwise. $\endgroup$
    – S. Carnahan ♦
    Oct 23, 2014 at 8:17
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    $\begingroup$ Nah, that's a theorem by now! A conjecture is an unproved statement with a lot of evidence in its favor ;) $\endgroup$
    – fedja
    Oct 23, 2014 at 10:04
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    $\begingroup$ This question needs a more descriptive title and better tags. $\endgroup$
    – Daniel Moskovich
    Oct 23, 2014 at 13:09
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    $\begingroup$ Strange question. Take $a=10^{12}$ and $f(x,y)=x^{\frac 13}-y^{\frac 13}$. Huxley's version of the prime number theorem in short intervals, gives the inequality for all large $x$ and $y$. Not clear to me why this has any more interest than Huxley's theorem! $\endgroup$
    – Lucia
    Oct 26, 2014 at 14:17
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    $\begingroup$ @Lucia: Please note that I am not looking for theorems that is of the form I mentioned. I am looking for unsolved conjectures which is of the form I mentioned. $\endgroup$
    – user57432
    Oct 27, 2014 at 11:39

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