Original examples of functions of slow increase in the spirit of Jakimczuk

Question

I believe that it is possible to prove that $$f(x)=e^{\operatorname{Ai}(x)}\log x$$ is a function of slow increase in the spirit of the definition given by the author of [1], where $\operatorname{Ai}(x)$ denotes the Airy function. Now I don't know if my example follows easily from the theorems of the article [1], or it is from the definition.

With independence of the applications to integer sequences, I think that these functions are interesting also in real analysis and were studied in the first section of the article, with consequences as Theorem 6 or other limits.

Question. Is it possible to show other different and original example of function of slow increase in the spirit of the definition due to Jakimczuk, and as a consequence of it an unexpected claim illustrating such example? Many thanks.

Thus I am asking about , if possible, other special functions or techniques to create unexpected functions that fits the cited definition, and if it is possible to provide some original consequence for this new example.

References:

[1] Rafael Jakimczuk, Functions of Slow Increase and Integer Sequences, Journal of Integer Sequences, Vol. 13, Article 10.1.1 (2010).

Answer 1

if $f(x)=e^{\operatorname{Ai}(x)}\log x$, then (because $\operatorname{Ai}(x)\rightarrow (4\pi)^{-1/2}x^{-1/4}e^{-\frac{2}{3} x^{3/2}}$ for $x\rightarrow\infty$) $$g(x)=\frac{xf'(x)}{f(x)}\rightarrow\frac{1}{\log (x)}$$ for $x\rightarrow\infty$, so indeed $g(x)\rightarrow 0$ and therefore $f(x)$ is "slowly increasing" as defined by R. Jakimczuk.