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Topic: Is it possible to bound these functions?
Replies: 3   Last Post: Apr 5, 2013 10:15 PM

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 Brad Cooper Posts: 167 Registered: 12/8/04
Is it possible to bound these functions?
Posted: Apr 4, 2013 9:21 PM

Define $A\{f(x)\}$ as a mapping from the set of functions defined on the interval $[0,1]$ to the Reals. \\

The functions are as "nice, smooth and integrable" as you may want them to be.

\begin{equation*}
A\{f(x)\} = {\left[\int_0^1 \cos\left(\int_0^x f(t)dt\right) dx\right]}^2 +
{\left[\int_0^1 \sin\left(\int_0^x f(t)dt\right) dx\right]}^2
\end{equation*}

Given that $a \leq f(x) \leq b$, can it be shown that $A\{a\} \geq A\{f(x)\} \geq A\{b\}$ ?

Cheers,