# Thread: Problem from Analisis 3 exam.

1. ## Problem from Analisis 3 exam.

Could there be function:

$\displaystyle f0,\infty) \rightarrow C$ so true: $\displaystyle \frac{1}{f} \in L^1(0,\infty)$ and $\displaystyle \overline{lim}_{t\rightarrow 0+} \frac{\int_0^t |f(x)|dx}{t^2}$ ?

Do I need to use some of inequalities or in any other way?

Thank you.

2. Originally Posted by veljko
Could there be function:

$\displaystyle f0,\infty) \rightarrow C$ so true: $\displaystyle \frac{1}{f} \in L^1(0,\infty)$ and $\displaystyle \overline{lim}_{t\rightarrow 0+} \frac{\int_0^t |f(x)|dx}{t^2}$ ?

Do I need to use some of inequalities or in any other way?

Thank you.
You have likely gotten no responses since this is incomprehensible. Try rewriting it. What is $\displaystyle L^1(0,\infty)$ Do you mean the space with the $\displaystyle \ell_1$ norm $\displaystyle \|(x_1,\cdots,x_n)\|=\sum_{j=1}^{n}|x_j|$?

3. Update:

Is there a function:

$\displaystyle f0,\infty) \rightarrow C$ so true: $\displaystyle \frac{1}{f} \in L^1(0,\infty)$ and $\displaystyle \overline{lim}_{t\rightarrow 0+} \frac{\int_0^t |f(x)|dx}{t^2} < \infty$?

p.s. $\displaystyle = \{f: \int_{(0,\infty)}|f|dx < + \infty \}$

4. edit:nevermind.
edit2: there's probably a one-liner involving maximal functions.

5. Originally Posted by veljko
Update:

Is there a function:

$\displaystyle f0,\infty) \rightarrow C$ so true: $\displaystyle \frac{1}{f} \in L^1(0,\infty)$ and $\displaystyle \overline{lim}_{t\rightarrow 0+} \frac{\int_0^t |f(x)|dx}{t^2} < \infty$?

p.s. $\displaystyle = \{f: \int_{(0,\infty)}|f|dx < + \infty \}$
How about $\displaystyle f(t)=e^t$.

6. Doesn't $\displaystyle \frac1{t^2}\int_0^te^x = \frac{e^t-1}{t^2}$ which goes to infinity?....