# Asymptotics of the derivatives of analytic functions

Are there sources that treat questions like the following ones?

1. Suppose that $f\colon\mathbb{C}\to\mathbb{C}$ is an entire function such that $f(x)$ is real for all real $x$ and $f(x)\sim1/x$ as $x\to\infty$. Does it then follow that $f^{(k)}(x)=O(1/x^{k+1})$ for $x>x_0$, where $x_0$ is a nonnegative real number?

2. Suppose that $f\colon\mathbb{C}\to\overline{\mathbb{C}}$ is a meromorphic function such that $f(x)$ is real for all real $x$ and $f(x)\sim x$ as $x\to\infty$. Does it then follow that $f^{(k)}(x)=O(x^{1-k})$ for $x>x_0$, where $x_0$ is a nonnegative real number?

In both cases, it would be good to also have more or less explicit bounds on the constants in $O(\cdot)$.

If the answer to such a question is negative in general, what additional conditions are needed to ensure the desired asymptotics?

• Consider what happens, for instance, if you change $f(z)$ to $f(z)+z^{-2}\cos({\pi/2}(e^z-z))$. – Michael Renardy Nov 5 '15 at 23:00
• Thank you Michael for the nice point. I admit that my question was not sufficiently well thought out; sorry. Pietro's reference to Carleman's result is quite educational to me. Still, is there a way to answer the more general question posed above: What additional conditions are needed to ensure the desired asymptotics? – Iosif Pinelis Nov 6 '15 at 4:47
• If the asymptotics applies in a sector rather than just the real axis, you can use Cauchy's formula for derivatives. – Michael Renardy Nov 6 '15 at 7:50
• Thank you Michael for another nice comment. I had indeed tried to bound $f^{(k)}(x)$ for "my" particular function $f$ of interest by showing that $|f(z)|=O(|z|+1)$ for $z$ in an open sector containing $(0,\infty)\subset\mathbb{R}$, but have succeeded only now in that. – Iosif Pinelis Nov 8 '15 at 2:25

I think the answer to both is negative. By a result by Carleman, entire real functions are dense in $C(\mathbb{R},\mathbb{R})$ with the Withney topology, so there is an entire $f$ such that $$1/x + \sin(e^x) /x^2 < f(x) < 1/x + \sin(e^x)/x^2 +1/x^3$$ for all $x>1$; this clearly forces $f'(x)$ to be unbounded for $x\to+\infty$. Analogously for the other question.
• You need to modify this slightly: your inequalities force $f(x) \to \infty$ as $x \to 0+$, which is not good for an entire function. Maybe make it $x > 1$. – Robert Israel Nov 6 '15 at 0:30
The answer to the second question is negative: Consider $$f(z) = \frac{z}{1+z^2}(1+\frac12 \sin (z^2))$$ Here the first derivative is asymptotic to 1, though the funtion goes to $\frac{1}{x}$.
• The first derivative oscillates forever in $(-1,1)$. – Brendan McKay Nov 6 '15 at 8:31