Show the inequality $\log(k)\log(n-k)<\log^2(n/2)$ holds

Question

How to show $\log(k)\log(n-k)<\log^2(n/2)$, for all $n,k >3$, where log is the natural log.

I plotted the function $\log(k)\log(n-k)-\log^2(n/2)$ and found the values are all negative, but I don't know how to show it.

Answer 1

Using AM-GM and Jensen's inequalities: $$\sqrt{\log k \cdot \log (n-k)} \le \frac{\log k + \log (n-k)}2 \le \log\frac{n}2$$ Equality is possible when $k = \dfrac{n}2$, so the inequality should not be strict.