Show that $$|\phi^{\pi}-\pi^{\phi}|\leq \operatorname{T},$$ where $\phi$ is the golden ratio and $\operatorname{T}$ the Tribonacci Constant
Using a calculator, we have $|\phi^{\pi}-\pi^{\phi}|=1.839144\cdots$ and $\operatorname{T}=1.83929\cdots$.
I have tried to make the problem more general using functions, but I would like to know if there are tricks to solve this problem.