> Almost but not quite linear in shape.

Seems like the intended meaning based on context.

https://en.m.wiktionary.org/wiki/sublinear

O(n^k) where 0 < k < 1

Would O(ln n) be "sublinear", though?

Of course. kuprel already said so.

I'm bad at math, but this is because d/dx ln(x) = x^-1, while d/dx x^k = kx^(-1+k), so as long as k > 0 the derivative of the latter is larger, right?

Well, the question here was just "is ln(x) sublinear or not?". You can answer that question with much less work: the second derivative is always negative, so the function must be sublinear. Any function that grows linearly must have a second derivative of zero.

Well sure but you can also just graph them or compare Taylor series (in radius of convergence, appropriate cut, etc)

More like o(n).

"Almost in a line."