The worst case height of AVL tree is $1.44 \log n$. How do we prove that? I read somewhere about Fibonacci quicks but did not understand it.
Try and create the "worst" AVL tree, that is, the AVL tree in which the height is as large as possible, with as least nodes as possible.
Once you get to the Fibonacci sequence (the answer to the question above), you can use the fact the the $n$-th Fibonacci number is $O(\phi ^n)$ (where $\phi = 1.618\ldots$), it should be easy to conclude that the worst case height of the tree is $\log_\phi n = O(\log n)$.