Is the following function convex? \begin{align} f(\alpha, \beta) = \exp\left( -j \cdot \alpha \cdot d^{\beta} \right), \end{align} where $j = \sqrt{-1}$, $\alpha \geq 0$, $\beta \geq 0$, and $d \in \mathbb{R}$.
If yes, how to show it?
If the function was a single variable dependent and twice differentiable, then one can show that the $f^{\prime\prime} \geq 0$ or show that $f(\lambda x_1 + (1-\lambda)x_2) \leq \lambda f(x_1) + (1-\lambda) f(x_2)$ is true.
But here a function of two variables $\alpha$ and $\beta$ is difficult for me. Can you please help me? Thank you so much in advance.
I think you can't define a convex function unless it takes real values. But there is a way to get around that by trying to prove that :
g(α,β):= |exp(−j⋅α⋅d^β)|is convex with the following definition :
g(λx1+(1−λ)x2)≤λg(x1)+(1−λ)g(x2) with x1 = (α1,β1) and x2 having the appropriate coordinate.
You can generalize the same line of thought if your function goes into any vector space, you introduce a norm.
First post here, make sure to tell me if i didn't awnser properly.