simple but (seemingly?) tricky recurrence [closed]

I came across the following very simple recurrence-style expression but am having trouble solving it: $$T(2n) \in \theta(T(n) \log(T(n)))$$ for sufficiently large $n \in \mathbb{N}$.

My first thought was to take the logarithm of both sides and apply the Master Theorem but the "$f(n)$" term unfortunately is not in the right form. Repeated expanding quickly yields a mess. Wolfram Alpha was no use.

Plugging in $T(n) = n^a$ makes the left side grow too slowly so $T$ must grow faster than any polynomial. But plugging in $T(n) = \exp(\log(n)^b)$, $b>1$, causes the left side to grow too fast so $T$ must grow more slowly. So it seems $T$ is super-polynomial but barely.

What approaches are viable for such an equation?

Edit: cross-posted here.

• Then why wouldn't you plug something barely super-polynomial, like $\exp(\log n\cdot\log\log n)$? Dec 25, 2017 at 9:17
• @IvanNeretin I have tried that function, and in the cross post, you can see that someone answered the question with that function. However, it isn't quite right. Dec 25, 2017 at 16:20
• Yes, I know, but at least this is a step in the right direction. Dec 25, 2017 at 16:22
• Please do not post the same question on multiple sites. Each community should have an honest shot at answering without anybody's time being wasted.
– D.W.
Dec 26, 2017 at 17:32
• I'm closing this question because further answers should better be posted on Computer Science now that it has a first answer there. Dec 26, 2017 at 19:09