Proving a given function is convex function

Question

I need help to prove the following question. Prove that function $f$ is convex on set $E \in R^n$ if and only if $$f(y + \beta(y - x)) \geq f(y) + \beta(f(y)-f(x))$$ for any $x, y \in E, \beta \geq 0$ and $y + \beta(y -x) \in E.$

Answer 1

See that you can re-write $y=\big(\frac{1+\beta}{1+\beta}\big)y+\big(\frac{\beta}{1+\beta}\big)x-\big(\frac{\beta}{1+\beta}\big)x=\big(\frac{\beta x}{1+\beta}+\frac{y}{1+\beta}+\frac{\beta}{1+\beta}(y-x)\big) $.
Then, $f(y)\leq \frac{\beta}{1+\beta}f(x)+\frac{1}{1+\beta}f(y+\beta(y-x))$ (by definition of convexity).
$\implies (1+\beta)f(y)-\beta f(x)\leq f(y+\beta(y-x))$. Hence, proved.