Proof that $\binom{2\phi(r)}{\phi(r)+1} \geq 2^{\phi(r)}$ I try to prove the following
$$\binom{2\phi(r)}{\phi(r)+1} \geq 2^{\phi(r)}$$
with $r \geq 3$ and $r \in \mathbb{P}$. Do I have to make in induction over $r$ or any better ideas?
Any help is appreciated.

Let $x_n=2^{-n}{2n\choose n+1}$. Then $x_{n+1}=x_n\frac{(2n+1)(n+1)}{n(n+2)}\gt x_n$ for every $n\geqslant1$ and $x_2=1$, hence $x_n\geqslant1$ for every $n\geqslant2$. 
Since $\varphi(r)\geqslant2$ for every integer $r\geqslant3$, the estimate above implies that the desired inequality holds for every (not necessarily prime) integer $r\geqslant3$.
