A polynomial approval

Question

How can I prove this expression mathematically? $$\prod_{i=1}^n (\theta x_i+(1-\theta)y_i)^{p_i} \ge{\theta {\prod_{i=1}^n} x_i^{p_i}} +(1-\theta){\prod_{i=1}^n}y_i^{p_i}$$ for $$ 0\le\theta\le1 \quad ,\quad 0\le{p_i}\le1 \quad ,\quad\sum_1^n {p_i}=1\quad ,\quad \forall x_i,y_i\in \Re \quad ,\quad x_i,y_i\gt0 $$ to me it is evident but I don't know how to prove it. Thank you!

Answer 1

Let $f(x)=x^p$ for $0 \leq p \leq 1$ and $x \geq 0$. Then by Jensen's inequality, for all $0 \leq \theta \leq 1$ and $x,y \geq 0$: $$ f(\theta x+(1-\theta)y) \geq \theta f(x)+(1-\theta)f(y)$$ since $f$ is concave. The inequality follows by multiplying out, ignoring all terms with $x_i y_j$ in them (since they are non-negative - reasonably, all $x_i$ and $y_i$ must be in this context).