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egasel
New poster

Post Number: 1
Posted on Wednesday, 30 March, 2011 - 09:05 am:   

Show that if [inline]F(x)[/inline] and [inline]G(x)[/inline] are continuous, non-negative and real-valued functions on the closed interval [inline][a,b][/inline], then:

[display] \int_a^b F(x) G(x) dx \leqslant \left ( \int_a^b [F(x)]^3 dx \right )^{\frac{1}{3}} \left( \int_a^b [G(x)]^{\frac{3}{2}} dx \right )^{\frac{2}{3}} [/display]

I have thought of using Cauchy-Schwarz inequality but dunno how it should be applied. Help!
Clare Fanthorpe
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Post Number: 6
Posted on Wednesday, 30 March, 2011 - 07:27 pm:   

What about if you write [inline]\int[F(x)]^3\textrm{d}x[/inline] as [inline]\int(F(x))[F(x)]^2\textrm{d}x[/inline] or as [inline]\int(F(x))(F(x))(F(x))\textrm{d}x[/inline] and use Cauchy-Scharz on that?
Luke Betts
Poster

Post Number: 11
Posted on Saturday, 02 April, 2011 - 11:27 am:   

This is really interesting -- it's Hoelder's inequality (which I've only seen for sums) done with integrals. I find it rather amusing how often you can take a theorem, replace all sums with integrals and still get something true. (Of course, the interesting case is where this isn't true.)

This suggests that if we can quote the sum form of Hoelder's inequality, then we can find some way to "take limits" to get our desired integral formulation (this comes with no guarantees -- I have not done it myself).

P.S. I suspect Clare's method will be `better', I just suggest this as an interesting (?) alternative.
lebesgue
Veteran poster

Post Number: 2101
Posted on Saturday, 02 April, 2011 - 11:35 am:   

Hoelder inequality is one of basic tools in measure and integration theory at various levels of generality, see wikipedia for more general account. In fact, the similar results for series or even for finite sums are just particular instances of more general theory with proper choice of the 'measure space'.
egasel
New poster

Post Number: 2
Posted on Wednesday, 04 May, 2011 - 03:38 pm:   

Thank you so much! This has been really helpful and interesting.

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