Any help with this prof is most wellcome.
Prove that, for any continuosly differentiable function f on [-phi, phi],
│ ∫[-phi, phi] f(t)cos(t) –f´(t) sin (t) dt│
≤ (2*phi)^½ * ({∫│f(t)^2 +│f´(t)│^2})^½
mathstudentdk@yahoo.com
Any help with this prof is most wellcome.
Prove that, for any continuosly differentiable function f on [-phi, phi],
│ ∫[-phi, phi] f(t)cos(t) –f´(t) sin (t) dt│
≤ (2*phi)^½ * ({∫│f(t)^2 +│f´(t)│^2})^½
mathstudentdk@yahoo.com
I think I have the solution. I'm going to use Hoelder inequality
$\displaystyle \int_S \bigl| f(x)g(x)\bigr| \,\mathrm{d}x \le\biggl(\int_S |f(x)|^p\,\mathrm{d}x \biggr)^{\!1/p\;} \biggl(\int_S |g(x)|^q\,\mathrm{d}x\biggr)^{\!1/q} $
Where $\displaystyle \frac{1}{p} + \frac{1}{q} = 1$
We see that
$\displaystyle f(t)\cos t - f'(t)\sin t = \mathrm{Re}\bigl[(f(t) + if'(t))(\cos t + i \sin t)\bigr]$
Hence
$\displaystyle I = \left|\int_{-\phi}^{\phi}\bigl(f(t)\cos t - f'(t)\sin t\bigr) dt \right|= \left|\mathrm{Re} \left[ \int_{-\phi}^{\phi}\bigl(f(t) + if'(t)\bigr) e^{it}dt \right] \right| \leq $
$\displaystyle \leq \left| \int_{-\phi}^{\phi}\bigl(f(t) + if'(t)\bigr) e^{it}dt \right| \leq \int_{-\phi}^{\phi}\bigl|f(t) + if'(t)\bigr|dt$
By Hoelder inequality we get
$\displaystyle I \leq \int_{-\phi}^{\phi}\bigl|f(t) + if'(t)\bigr|dt \leq \sqrt{\int_{-\phi}^{\phi}dt} \sqrt{\int_{-\phi}^{\phi}\bigl|f(t) + if'(t)\bigr|^2dt}$
Which, you can easily see implies the inequality we were to prove