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stilyo
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Joined: January 12th, 2009, 6:31 pm

Integral Inequality

June 20th, 2018, 7:10 pm

Let A = Integral [0...inf] f(t)dt, A>0.
Suppose I have the following integral: B=Integral [0...inf] f(t)*exp(-kt)dt where k>0 is a constant. Is it possible to have a bound for B (upper or lower) in terms of A and k? I tried Cauchy-Schwarz but that only gets you to integral of f(t)^2 and not sure if I can express this in terms of A somehow...
 
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ppauper
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Re: Integral Inequality

June 20th, 2018, 9:53 pm

if it helps any, those are Laplace transforms
[$]B=F(k)=\int_{0}^{\infty}f(t)e^{-kt}dt[$]
[$]A=F(0)[$]

are there any conditions on [$]f(t)[$] ?
 
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Cuchulainn
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Re: Integral Inequality

June 21st, 2018, 11:14 am

A wild guess is that this is the Laplace transform of a linear (1st order?) ordinary differential equation and you want to bound the solution in terms of its input data?

As ppauper states, the function [$]f(t)[$] should be qualified (e.g. is it of exponential order etc.?)

If you can pose it as an ODE then you can conclude a desired positivity/maximum principle. Maybe there isn't a corresponding ODE and then I don't know.