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$L,M,N$ are three positive integers such that $1 \le M, M \le N, M \le L$ and $M$ is a divisor of $N$. It appears that the following inequality is correct (after assigning many random values to $M,N,L$, such that the conditions above are satisfied), but I was not able to prove it:

$${\left( {1 - \frac{M}{L}} \right)^{N/M}} \le {\left( {1 - \frac{1}{L}} \right)^N}$$

Any help would be greatly appreciated.

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This is equivalent to $(1-1/L)^M \ge 1-M/L$ which can be proved by induction on $M$. Moreover this is true for any real $L>1$ and $N$ has no effect here at all. –  njguliyev Oct 3 '13 at 14:11
    
Thanks! the inequality you wrote is essentially Bernoulli's inequality. –  rc1 Oct 3 '13 at 16:03

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up vote 1 down vote accepted

As you mentioned in one comment yourself, this is an instance of Bernoulli inequality which is as follows:

Suppose $x>-1$ and $y\in (0,1)$, then $(1+x)^y\leq 1+xy$.

It is not difficult to see that the above inequality followis by choosing $x=\frac{-M}{L}$ and $y=\frac{1}{M}$ in Bernoulli inequality.

However it is more interesting to see the proof of Bernoulli inequality itself. To do that we use Jensen inequality. $\ln(x)$ is concave and therefore we have: $$ y\ln(1+x)+(1-y)\ln 1 \leq \ln\left(y(1+x)+(1-y).1\right)=\ln(1+xy). $$

which gives the proof of Bernoulli inequality.


For the case of $L=M=1$, we have the equality. Bernoulli inequality does not apply here.

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