Let
$$\varphi(x) = \begin{cases} -\frac{1}{p}x^p &, \mbox{ if } x \geq 0, \mbox{ where } 0 < p < 1, \\ \infty &, \mbox{ if } x < 0. \end{cases}$$
I want to show convexity of this function. This is my approach so far:
let us assume $x, y > 0$. Given $t \in (0, 1)$, we have $$ \varphi(tx + (1 - t)y) = -\frac{1}{p}(tx + (1 - t)y)^p $$ and $$ t\varphi(x) + (1 - t) \varphi(y) = -\frac{t}{p} x^p - \frac{(1 - t)}{p}y^p = -\frac{1}{p}(tx^p + (1 - t) y^p). $$ Now we show $$ (tx + (1 - t)y)^p \geq (tx^p + (1 - t)y^p). $$ Then we have $$ tx + (1 - t)y \geq (tx^p + (1 - t)y^p)^{\frac{1}{p}} $$ I am not sure how to show this. It seems like Holder inequality, but I am not sure how one would apply it here. I am looking for an approach without using derivatives. Any advice is appreciated.