Let $$f(z) = (1-1/t) z^w + z/t - 1$$ with integers $t\geq2$ and $w\geq2$.Let $r=1+1/(tw^3)$. How do I show $$\left\lvert f(r e^{i\varphi}) \right\rvert \geq \left\lvert f(r) \right\rvert$$ for any $\varphi$?

It is clear, that there is a local minimum at $\varphi=0$. The inequality is relatively easy to prove for $w=2$.

The problem is equivalent to show, that $$\cos(\varphi)-1 + (1-1/t) r^w \left(\frac{t}{r} \left(\cos(w\varphi)-1\right) - \left(\cos((w-1)\varphi)-1\right) \right) \leq 0.$$ This formulation comes from writing the square of the absolute value in terms of sin and cos.

How can I prove this?