Prove that the second derivative is positive iff the function is convex.

Question

Well, I want to prove the following:

Let $f:(a,b)\to\mathbb R$ be double differentiable then $f$ is convex iff $\;\;f''(x)>0$ for all $x\in (a,b)$.

Then I tried te following:

$\Rightarrow]$ Lets suppose that $f:(a,b)\to\mathbb R$ is convex then if we define $h=-y+x$ we have that:

$$ f(x+h) \leq (1-t)f(x)+tf(x+h)$$

$$0 \leq (1-t)f(x)+tf(x+h)-f(x+h)$$

I was expecting to have something of this form

$$0 \leq f(x+h)-2f(x)+f(x-h)$$

and then take limit and have the result, I don't know how to define $h$ to have the above relation, Can someone help me ?

$\Leftarrow]$ I checked this, but I don't know if it is right, If it is not Can you help me to fix the mistakes please:

Thanks a lot in advance

score 4 · Answer 1 · answered Mar 04 '15 at 18:57

4

From convexity, $f(ta+ (1-t)b) \leqslant tf(a) + (1-t)f(b)$.

Let $t = 1/2$, $a= x-h$, and $b = x+h$.

Then

$$f(x) \leqslant \frac1{2}f(x-h) + \frac1{2}f(x+h)\\ \implies f(x+h) - 2f(x) + f(x-h) \geqslant 0$$

answered Mar 04 '15 at 18:57

RRL

Thanks a lot s a lovely proof, but can I fix t? – user162343 Mar 04 '15 at 18:58
Convexity means the inequality holds for any $t \in [0,1]$ – RRL Mar 04 '15 at 18:59
Right :) thanks a lot, what about the second part? – user162343 Mar 04 '15 at 19:00
Is that proof right? – user162343 Mar 04 '15 at 19:00
Are you also trying to show $f''(x) > 0 \implies $ convexity. Use Taylors Theorem. – RRL Mar 04 '15 at 19:00
Yes, thats right but There is a discussion in the above link about it and I want to be sure that is right :) – user162343 Mar 04 '15 at 19:02
@user162343: The proof in the linked question is correct. Multiply the first and second inequalities by $\lambda$ and $(1-\lambda)$, respectively, and add. The terms involving $f'$ cancel and you get the convextity condition. – RRL Mar 04 '15 at 19:42
Ok then with that proof the second part is covered right? – user162343 Mar 04 '15 at 19:43
Yes, $f'' >0 \implies$ convexity – RRL Mar 04 '15 at 19:44
ok thanks a lot ;), only one thing I want to prove that $a^{1-t}>(1-t)a+tb$, then I should prove that $t \to a(b/a)^{t}$ is convex but how can I define the function to apply the criterion ? – user162343 Mar 04 '15 at 19:49
What Im trying to ask is how can I define an $f$ for the relation $$t \to a(b/a)^{t}$$ – user162343 Mar 04 '15 at 19:54
@LucasMation: Please read this definition. – RRL Sep 19 '18 at 16:36

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