# 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 [1] (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, [2] (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:

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

[2] 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