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Recently, I have become intrigued with this functional:$$D_n=\lim \limits _{x\to \infty}\frac{f'(x)}{[f(x)]^n}.$$In particular, provided that the function is both differentiable and increasing in magnitude for all $x$, for which functions does $D_n$ diverge to infinity? First, I have considered $n=1$. This has obvious solutions. For example, $e^{e^x}$, which has derivative $e^x\cdot e^{e^x}$, makes $D_1$ diverge to infinity, but not $D_2$. Is there any function which satisfies the above conditions and makes $D_2$ diverge? What about a function which satisfies the above conditions and makes $D_n$ diverge for all values of $n$? As a reminder, here are the conditions for the functions:

  • You must be able to choose an $x_0$ such that the function is differentiable for all $x>x_0$.
  • The absolute value of the function must be increasing for all $x$ on which it is defined.
  • The function must be both defined on real numbers and real-valued.
commie trivial
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Toby Saunders-A'Court
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    Here's a thought, but not a proof. If the functions were continuously increasing, etc, then they would be bounded below by the function that satisfies the ODE $$f' = kf^n$$ for $k > 0$ by Gronwall inequality. But the solutions to this ODE $$f(x) = \frac{1}{\sqrt[n-1]{x_f - kx}}$$ all blow up in finite time, so the even greater functions must diverge even faster and have limited intervals of convergence as well (which violates your condition 1). This brings functions like the infinite power tower on the table, though. – Ninad Munshi May 16 '22 at 23:40

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