Identify the regions in $(\alpha_{1},\ \alpha_{2})$ plane for which $f_{\alpha_{1},\ \alpha_{2}}$ is a convex or concave function

Question

A family of functions. For constants $\alpha_{1}$ $\alpha_{2}$,

let $f_{\alpha_{1},\alpha_{2}}:R^{2}_{+} \rightarrow R_{+}$ be defined as:

$\hspace{4 cm}$ $f_{\alpha_{1},\alpha_{2}}(x_{1},\ x_{2})= x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}, \hspace{2 cm} x_{i} \geq 0$

Identify and draw the regions in the $(\alpha_{1},\ \alpha_{2})$ plane for which $f_{\alpha_{1},\ \alpha_{2}}$ is a

(a) convex function

(b) concave function

Advance phase: Generalize the previous result to this case:

for a vector $\alpha \in R^{n},$ let $f_{\alpha}:R^{n}_{+} \rightarrow R_{+}$ be defined as:

$\hspace{4 cm}$ $f_{\alpha}(x)=\displaystyle\prod_{i=1} ^{n} x_{i}^{\alpha_{i}}, \hspace{2 cm} x_{i} \geq 0$

Any help would be appreciated.

Thanks very much in advance.

Answer 1

Hint: There is first and second order characterizations of convex functions here

Well known result is following: Suppose function $f$ is defined and have continuous second partial derivatives in interior of convex set $X \subset \mathbb{R}^n$. $f$ is convex if and only if when quadratic form $$\sum_{i,j=1}^{n}f_{ij}(x)\xi_i \xi_j$$ is positive semidefinite.