Suppose $p<1,p\ne0$. Show that the function $$ f(x) = \left(\sum_{i=1}^{n} x_i^p \right)^{\frac{1}{p}} $$ with dom$\ f=\Bbb{R}_{++}^n$ is concave, where dom represents domain and $\Bbb{R}_{++}=\{x\in\Bbb{R}:x>0\}$. Hint. Adapt the proofs for the log-sum-exp function and the geometric mean.
- Log-sum-exp.$\quad$The function $f(x)=\log(e^{x_1}+\cdots+e^{x_n})$ is convex on $\Bbb{R}^n$.
- Geometric mean.$\quad$The geometric mean $f(x)=(\Pi_{i=1}^{n} x_i )^{1/n}$ is concave on dom$\ f=\Bbb{R}_{++}^n$.
I can prove whether $f(x)$ is concave or not using 2nd derivative of the function. However, I have failed to prove it using the above two things; log-sum-exp and geometric mean. Please let me know how to solve it? Thank you for reading my question.