# Is convergent $\int_0^{+\infty}\sin\log\left(\frac{f(x)}{xf'(x)}\right)dx$, if $f(x)$ is of slow increase in spirit of Jakimczuk's definition?

Let $f(x)$ of slow increase thus satisfying all requirements of Jakimczuk's Definition 1 from  (it is a free access journal).

I wondered next question as a combination of previous requirements and definition and PROBLEMA 151, proposed in LA GACETA,  (in Spanish).

Question. Can you prove that if being $f(x)$ a function satisfying Jakimczuk's definiton of functions of slow increase then the integral $$\int_0^{+\infty}\sin\log\left(\frac{f(x)}{xf'(x)}\right)dx\tag{1}$$ is convergent? Many thanks.

Thus I think (I think that my question has mathematical meaning) that I am asking about if you can find a counterexample of a function that is of slow increase but being such that $(1)$ doesn't converge, or well that you can to provide me a proof of the statement written in previous question: Let $f(x)$ a function being of slow increase (with Jakimczuk's requirements and definition). Then the integral $(1)$ is convergent.

## References:

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

 PROBLEMA 151, proposed by Paolo Perfetti, La Gaceta de La Real Sociedad Matemática Española, Vol. 14, nº 2, p.285. The link PROBLEMAS Y SOLUCIONES, here.

• Have you tried to investigate that for a simple example of a function of slow increase? Do you have any reason to think the statement may be true? – Professor Vector Dec 17 '17 at 18:14
• Please make this question self contained and include Jakimczuk's definition – Brevan Ellefsen Dec 17 '17 at 18:48
• Many thanks @BrevanEllefsen to you and Professor Vector. I am going to think what should do I with this question. – user243301 Dec 17 '17 at 19:59