# functions – Proving Jensen’s Inequality by NOT using induction.

Let $$f:Irightarrow mathbb R$$ be a convex function. Show that for any integer $$n$$, any real numbers $$x_1,x_2,ldots, x_nin I$$ and any positive real numbers $$mu_1, mu_2,ldots, mu_n$$ such that $$sum_{i=1}^{n}mu_i=1:$$
$$sum_{i=1}^{n}mu_if(x_i)geq fbigg(sum_{i=1}^{n}mu_ix_ibigg)$$

Now I specifically want to prove it WITHOUT using induction but I’m stuck at a point. Here’s what I’ve tried:

We know that: $$sum_{i=1}^{n}mu_if(x_i)geq mu_1f(x_1)+f(x_k)sum_{i=2}^{n}mu_ispace text{where f(x_k)=min{f(x_i)}space forall 1leq ileq n}$$

Since $$sum_{i=1}^{n}mu_i=1$$, we would have $$sum_{i=2}^{n}mu_i=1-mu_1$$

Now since $$f$$ is a convex function: $$mu_1f(x_1)+f(x_k)sum_{i=2}^{n}mu_i=mu_1x_1+(1-mu_1)f(x_k)geq f(mu_1x_1+(1-mu_1)x_k)$$

Thus: $$sum_{i=1}^{n}mu_if(x_i)geq f(mu_1x_1+x_ksum_{i=2}^{n}mu_i)$$

So if I prove that $$f(sum_{i=1}^{n}mu_ix_i)leq f(mu_1x_1+x_ksum_{i=2}^{n}mu_i)$$ (if it is true), then I’m done.

But this is the step where I’m stuck. Would you please help me prove this?

THANKS

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