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Let $f\left( x\right)$ be a $C^{2}$ function on $\mathbb{R}$. Show that $$\sup \left| f'\left( x\right) \right| ^{2}\leqslant4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| .$$

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3  
What have you tried? –  Matt Pressland Dec 12 '12 at 11:17
1  
This is not true. Take $f(x)=1$ for all $x\in\mathbb{R}$ which is $C^2$. The LHS is $1$, but RHS is $0$. –  Paul Dec 12 '12 at 11:25
5  
There is a typo. The LHS should be $|f'(x)|^2$. Rudin's PoMA has this as an exercise. –  lhf Dec 12 '12 at 11:41
2  
I downvoted. It is very annoying to see questions in the imperative like this. –  Giuseppe Negro Dec 12 '12 at 13:23

1 Answer 1

up vote 10 down vote accepted

Let $\sup|f^{(n)}(x)|=M_n$. Since $f$ is $C^2$-continuous, we can use Taylor's theorem to write:

$$f(x+h)=f(x)+hf'(x)+\frac{h^2}2f''(t)$$

for some $x<t<x+h$. Then we can solve for $f'(x)$:

$$f'(x)=\frac1h(f(x+h)-f(x))-\frac h2f''(t)$$ $$|f'(x)|\leq\frac1h(|f(x+h)|+|f(x)|)+\frac h2|f''(t)|\leq\frac2hM_0+\frac h2M_2.$$

Since we can apply this for any $h$, choose $h=2\sqrt{M_0/M_2}$, so that

$$|f'(x)|\leq\sqrt{M_2/M_0}\cdot M_0+\sqrt{M_0/M_2}\cdot M_2=2\sqrt{M_0M_2}.$$

Since this inequality is true for any $f'(x)$, it is true for the supremum, i.e.

$$M_1\leq2\sqrt{M_0M_2}\Rightarrow\sup|f'(x)|^2\leq4\sup|f(x)|\sup|f''(x)|.$$

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2  
+1 Nice! I'm linking an old post of mine which uses essentially the same technique, for reference: math.stackexchange.com/a/44538/8157 –  Giuseppe Negro Dec 12 '12 at 13:22
    
Sweet solution! I think you have typo in the second last equation. It should be $+$ instead of $-$ between the terms. –  Prism Jul 24 '13 at 20:28
    
@Prism You're right; I fixed it. –  Mario Carneiro Jul 25 '13 at 5:12

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