关于单位圆上函数族$f_k(\varphi)$实部全局最小值的证明问询

Hi everyone, I'm stuck on verifying a key numerical observation about a family of functions defined on the unit circle, and I'm hoping someone can share a hint or insight to help me prove it. Let me walk through the problem details:

Problem Setup #

Let $n \geq 3$, and consider the family of functions ${f_k}$ indexed by $k = 1,2,\ldots,n-1$, defined on the unit circle as:
$$f_k(\varphi) = \textrm{Re}\left[e^{{-ik\varphi}\left(1-(1-e}{i\varphi})\frac{k}{n}\right)^n\right] = \sum_{j=0}^n {n\choose j} \left( \frac{k}{n} \right)^j \left( 1 - \frac{k}{n} \right)^{n-j} \cos(k-j)\varphi$$

Known Properties #

I've already worked out a few basic characteristics of these functions:

• Symmetry: $f_k(-\varphi) = f_k(\varphi) = f_{n-k}(\varphi)$
• Global maximum: $\max_{k,\varphi} f_k(\varphi) = 1$, which is attained at $\varphi = 0$ for any $k$

The Unproven Observation #

From numerical calculations, I've noticed that the global minimum of this function family is:
$$\min_{k,\varphi} f_k(\varphi) = -\left(1-\tfrac{2}{n}\right)^n$$
And this minimum is only achieved when $k \in {1,n-1}$ at $\varphi = \pm \pi$.

My Failed Attempt #

I tried finding critical points by solving $f_k'(\varphi) = 0$, but this led to an algebraic mess that I couldn't simplify into any useful conclusions. I'm stuck on figuring out a better approach to tackle this.

If anyone has a different angle to attack this proof, or can point me toward a relevant theorem or technique, I'd be really appreciative!

备注：内容来源于stack exchange，提问作者meler