I am writing an essay on the gamma function. I have learnt and understood convex theory and how the log-convex nature of the gamma function makes it a unique extension of the factorials (Bohr-Mollerup). However, I am still a little unsure as to the real intuition regarding log-convexity. I have read it as an indication of the extreme smoothness of a function if it is log-convex. But why is this? Is it because of the seemingly smooth nature of the $ln$ function?
Explanations would be appreciated!