Best known inequality for the larger prime number of a product?

Question

We all know given two prime number's $b$ and $a$ whose product is $c$:

$$ c \geq b \geq \sqrt c \geq a \geq 2 $$

where, $ab=c$

I was wondering if the inequality for $b$ could be improved upon and discovered:

$$ b \geq \lfloor\sqrt{5c + 4 - 4 \sqrt{c}} - \sqrt c \rfloor $$

where $\lfloor x \rfloor$ is the floor function.

This made me wonder what is the best known inequality for the larger prime number $b$?

Answer 1

The best bound is $$b \geq \sqrt{c}$$

This is because equality can always hold in the case that $c$ is the square of a prime number, since $$a=b=\sqrt{c}$$ This means that the inequality cannot be improved.