# Prove logarithmic inequality with greatest integer function.

$\left \lfloor n\log_2 n^2 \right \rfloor + \left \lfloor \log_2(\left \lfloor n\log_2n^2 \right \rfloor) \right \rfloor \leq \left \lfloor (n+1)\log_2 (n+1)^2 \right \rfloor + 1$

How to show this? I tried using $\left \lfloor k \right \rfloor \leq k$, but I'm not sure that that does anything...

LHS $\leq \left \lfloor (n+1)\log_2 (n+1)^2 \right \rfloor + \left \lfloor \log_2(\left \lfloor n\log_2n^2 \right \rfloor) \right \rfloor$ < $\left \lfloor (n+1)\log_2 (n+1)^2 \right \rfloor + 1$ because...ugh...

$\left \lfloor \log_2(\left \lfloor n\log_2n^2 \right \rfloor) \right \rfloor \leq \log_2(\left \lfloor n\log_2n^2 \right \rfloor) < 1$?

I don't think the last inequality is true.

I think it's supposed to hold for all natural numbers or at least all natural numbers after a certain index.

• Are you trying to prove this $\forall n \in \mathbb{N}$? Commented Aug 24, 2014 at 0:04
• @Phrohlych I think it's supposed to hold for all natural numbers or at least all natural numbers after a certain index. Edited into question. Thanks for clarification.
– BCLC
Commented Aug 24, 2014 at 0:36
• Okay. The smallest integer for which this makes sense is $2$, so that would be your starting point. Using induction, I was able to prove the base case (i.e. $n=2$). I haven't yet attempted the inductive step, but that might be the direction where you want to head. Commented Aug 24, 2014 at 0:38
• @Phrohlych Ayt. Will try that out. Thanks
– BCLC
Commented Aug 24, 2014 at 0:46

hint:

$\left \lfloor (n+1)\log_2 (n+1)^2 \right \rfloor + 1 =\left \lfloor (n\log_2 (n+1)^2)+\log_2 (n+1)^2 \right \rfloor +1\ge \left \lfloor (n\log_2 (n+1)^2) \right \rfloor +\left \lfloor \log_2 (n+1)^2\right \rfloor+1$

now prove:

$\left\lfloor (n\log_2 (n+1)^2) \right \rfloor \ge \left\lfloor (n\log_2 n^2) \right \rfloor$

$\left \lfloor \log_2 (n+1)^2\right \rfloor+1 \ge \left \lfloor \log_2(\left \lfloor n\log_2n^2 \right \rfloor) \right \rfloor$

• Sigh. I did not know or forgot that property of greatest integer function. Thanks.
– BCLC
Commented Aug 24, 2014 at 12:49