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?



Asymptotics of the derivatives of analytic functions
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.


