# Upper limit function for the decay of $\int_x^\infty f(y)dy$, provided $f(y)$ is analytic, when $x\rightarrow\infty$?

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?

• 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 $f(x) = 1/(1+x^2)$. – Robert Israel Oct 10 '16 at 4:17
$f(x):=\frac{\sin^2 x}{x^2}$ is entire, bounded on $\mathbb R$, and $\int_x^\infty f(y)\ dy$ decreases as $cx^{-1}$.
Even knowing that $f$ is entire would give no information at all. Check this Carleman's theorem.