Are there sources that treat questions like the following ones?

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?

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?