Question. Are the following conditions equivalent? \begin{align} \textbf{Condition 1}: \quad & f > 0 ~ \text{and} ~ \frac{\partial^{3} f}{\partial x^{3}} ~ \text{exists}. \\ \textbf{Condition 2}: \quad & \frac{\partial^{3} \log f}{\partial x^{3}} ~ \text{exists}. \end{align}
My answer:
Yes, first I show that Condition 2 implies Condition 1. Obviously, if $ f \leq 0 $, then $ \dfrac{\partial^{3} \log f}{\partial x^{3}} $ will not exist, therefore $ f > 0 $. Since $ f = e^{\log{f}} $, we can easily see that if $ \dfrac{\partial^{3} \log f}{\partial x^{3}} $ exists, then $ \dfrac{\partial^{3} f}{\partial x^{3}} $ exists.
Now, I show that Condition 1 implies Condition 2. By the chain rule of derivation (Faà di Bruno’s Formula), if all necessary derivatives are defined, then \begin{align} \frac{\partial^{3} \log f}{\partial x^{3}} & = \frac{\partial^{3} \log f}{\partial f^{3}} \left( \frac{\partial^{3} f}{\partial x^{3}} \right)^3 + 3 \frac{\partial^{2} \log f}{\partial f^{2}} \frac{\partial f}{\partial x} \frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial \log f}{\partial f} \frac{\partial^{3} f}{\partial x^{3}} \\ & = \frac{2}{f^{3}} \left( \frac{\partial^{3} f}{\partial x^{3}} \right)^{3} + 3 \frac{-1}{f^{2}}\frac{\partial f}{\partial x} \frac{\partial^{2} f}{\partial x^{2}} + \frac{1}{f}\frac{\partial^{3} f}{\partial x^{3}}. \end{align} Thus, Condition 1 implies Condition 2.
Please let me know if I am correct and if both conditions above are indeed equivalent.