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Question with three parts:

a. Is there a theorem which states an upper limit function for rate of the decay of $\int_x^\infty f(y)dy$, as $x\rightarrow\infty$?

b. Assume $f(y)$ is positive, real, and restriceted to the positive-real y's, bounded from above by some constant 'a' and $\lim_{y\rightarrow\infty}[f(y)]\rightarrow0$ and is analytic in $[0,a]$:

Is it always possible to say something like e.g., $\int_x^\infty f(y)dy\leq exp(-cx)$, when $x\rightarrow\infty$?

c. If the answer to a.&b. is "nope". What are the general properties this integral in this limit?

Thanks on advance!

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    $\begingroup$ It's not clear what the assumptions are supposed to be. Presumably you're assuming the improper integral exists. Is "$f(y)$ is positive and bounded above" restricted to the positive reals, but "everywhere analytic" means entire? Otherwise there are obvious counterexamples such as $f(x) = 1/(1+x^2)$. $\endgroup$ – Robert Israel Oct 10 '16 at 4:17
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$f(x):=\frac{\sin^2 x}{x^2}$ is entire, bounded on $\mathbb R$, and $\int_x^\infty f(y)\ dy$ decreases as $cx^{-1}$.

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Even knowing that $f$ is entire would give no information at all. Check this Carleman's theorem.

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  • $\begingroup$ Can you pls specify a bit more why the theorem says there're no general properties for this expression? Leibniz integral rule doesn't supply information? Maybe one can use it to maximize this integral? $\endgroup$ – user1611107 Oct 10 '16 at 15:48

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