Problema 2 ONM 2008

[Cezar Lupu]

Se considera functia \( f:[0,1]\to\mathbb{R} \), derivabila, cu derivata continua pe \( [0,1] \). Sa se arate ca daca \( f(1/2)=0 \), atunci

\( \int_0^1(f^{\prime}(x))^{2}dx\geq 12\left(\int_0^1f(x)dx\right)^{2} \).

Cezar Lupu, Tudorel Lupu

[o.m.]

Use Fubini theorem
\( \int_{0}^{1}f(x)dx=\int_{0}^{1}(\int_{1/2}^{x}f^{\prime}(t)dt)dx=\int_{0}^{1}k(t)f^{\prime}(t)dt \)

where \( k(t)=-t \) on \( [0;1/2] \) and \( k(t)=1-t \) on \( [1/2;1] \).

With Cauchy-Schwarz
\( (\int_{[0,1]}f)^2 \leq (\int_{[0;1]}k^2)(\int_{[0;1]}f^{\prime2}) \)

and
\( \int_{[0;1]}k^2=\int_{[0;1/2]}t^2dt+\int_{[1/2;1]}(1-t)^2dt=1/24+1/24=1/12 \)

[aleph]

Your inequality with "24" is valid only for small values of 24
More seriously, the second "=" is wrong.
The question if 12 is the best constant remains.

[Cezar Lupu]

Your inequality with "24" is valid only for small values of 24
More seriously, the second "=" is wrong.
The question if 12 is the best constant remains.

Yes, the best constant is 12. The problem is not that easy as it seems at first glance.
My solution (which is elementary) is based on Cauchy-Schwarz inequality in the following way:

\( \int_0^{1/2}(f^{\prime}(x))^{2}dx\cdot\int_0^{1/2}x^{2}dx\geq\left(\int_0^{1/2}xf^{\prime}(x)\right)^{2} \)
and
\( \int_{1/2}^{1}(f^{\prime}(x))^{2}dx\cdot\int_{1/2}^{1}(1-x)^{2}dx\geq\left(\int_{1/2}^{1}(1-x)f^{\prime}(x)\right)^{2} \).

By adding these two inequalities and using the elementary inequality \( \alpha^2+\beta^2\geq\frac{(\alpha+\beta)^{2}}{2} \) and integration by parts, we get the desired result. \( \qed \)