I was studying Inequalities when I came across Jensen Inequality
It is applied to convex function and the inequality is flipped when applied to concave functions
I read from different sources but I wasn't able to add up convex and concave functions
Can anyone help me out here ?
BTW if possible please present your answer in simple language ( and terms)
On
$f$ named a convex function if for all $a<b$ and for all $x\in[a,b]$ the point $C(x,f(x))$ is placed under point $D(x,y)$, where $D\in AB$, $A(a,f(a))$, $B(b,f(b)).$
If $f$ is a convex function then for all $\alpha_i\geq0$ such that $\alpha_1+\alpha_2+...+\alpha_n=1$ and for all $x_i$ we have: $$\alpha_1f(x_1)+\alpha_2f(x_2)+...+\alpha_nf(x_n)\geq f(\alpha_1x_1+\alpha_2x_2+...+\alpha_nx_n).$$
A function $f$ is convex if, given any two points $A \,(a, f(a))$ and $B\,(b,f(b))$ on the representative curve of $f$, the arc of curve from $A$ to $B$ is beneath the chord $[AB]$.? Parametrizing numbers between $a$ and $b$, we see that numerically, this means that $$\forall t, 0\le t\le 1,\enspace f(at+(1-t)b)\le tf(a)+(1-t)f(b).$$
As this parametrisation amounts to consider a number between $a$ and $b$ as a barycenter of $a$ and $b$ with positive weights, Jensen's inequality is just a generalisation to the case of a barycenter of $n$ numbers, and we can describe convex functions as ‘subbarycentric’, or ‘subaffine’, since affine functions are those functions which preserve barycenters.