Euler's totient function - inequality on even numbers?

Question

Is there a theorem that says:

If $n$ is even then: $$\varphi(n)\le\frac{n-1}{2},$$ where $\varphi(n)$ is Euler's totient function.

If there isn't such a theorem, how would I show/prove that this is true?

Answer 1

If you combine the two comments you'd have:

Case 1: $n=2^k m, k\ge 1$, $m>1$ odd, then $$\phi(n)=\phi(2^k)\phi(m) = 2^{k-1} \phi(m) \le 2^{k-1} (m-1) < 2^{k-1} m - \frac 12 = \frac{n-1}{ 2}$$

Case 2: If $m=1$ then $\phi(n)=2^{k-1} =\frac n2 > \frac{n-1}{2}$.