Conjecture :
Let $\phi(m)$ be Euler's totient function
$1 \leq \phi(m) \leq \lceil \frac{m-1}{2} \rceil ~~$ if $~~m~~$ is even
$\lceil \frac{m+1}{3} \rceil \leq\phi(m) \leq m-1 ~~$ if $~~m~~$ is odd
Proof :
Case when $ m$ is an even number :
Let us make substitution : $m=n-1$ , so:
$n=2^r \cdot q_1^{r_1} \cdot q_2^{r_2} \cdots q_k^{r_k}+1 \Rightarrow $
$\Rightarrow \phi(n-1)=\phi(2^r) \cdot \phi(q_1^{r_1})\cdots \phi(q_k^{r_k})=\phi(2^r) \cdot q_1^{r_1-1}(q_1-1)\cdots q_k^{r_k-1}(q_k-1)=$
$=\phi(2^r) \cdot q_1^{r_1} \frac{(q_1-1)}{q_1} \cdots q_k^{r_k} \frac{(q_k-1)}{q_k} < \phi(2^r) \cdot \frac{n-1}{2^r}=2^{r-1}\cdot \phi(2) \cdot \frac{n-1}{2^r}=\frac{n-1}{2}= \lceil \frac{n}{2}-1 \rceil \Rightarrow$
$\Rightarrow \phi(m) < \lceil \frac{m-1}{2} \rceil$
Question : How to prove the second part of the statement, (case when $m$ is an odd number) ?