On the integral  $\int_{b'}^{b''}f(x)\ln(1+a x)\mathrm{d}x$ If I  have a function $f(x)$ that is $C^2$ and I know that $\int_{-\infty}^{+\infty}f(x)\mathrm{d}x=1$ and $\int_{-\infty}^{+\infty}x f(x)\mathrm{d}x=\mu$ in real axis $x$, what can I say about this integration in the real axis 
$$
G(a)=\int_{b'}^{b''}f(x)\ln(1+a x)\mathrm{d}x
$$
knowing that $b',b'',a,\mu$ are real?
 A: The Taylor series expansion of $\log(1+ax)$ around the point $b'$ (assuming $b'$ is chosen in a valid range) is $\log(1+ab') + \frac{a}{1+ab'}(x-b')+O(x^{2})$. If we assume $b''$ is close enough that a linear approximation works and we plug this in then:
$$ \int_{b'}^{b''}f(x)\log(1+ax)dx$$ 
$$\approx \log(1+ab')\cdot\biggl(F(b'')-F(b')\biggr) + \frac{a}{1+ab'}\cdot{}\biggl(\int_{b'}^{b''}xf(x)dx - b'[F(b'')-F(b')]\biggr)$$
$$ = \biggl[ \log(1+ab') - \frac{ab'}{1+ab'}\biggr]\cdot{}\biggl(F(b'')-F(b') \biggr) + \frac{a}{1+ab'}\int_{b'}^{b''}xf(x)dx,$$
where $F(x)$ is the CDF coming from the known density $f(x)$. 
Like others have mentioned, you won't be able to squeeze much out of the last term without making significant assumptions of the form of $f(x)$ on $[b',b'']$. If you're willing to explore various assumptions about whether the random variable $X$ that has $f(x)$ as its density is bounded or non-negative, then you can probably make use of Bennett's inequality and/or Markov's inequality to get some inequality constraints.
Another approach, which I may try to flesh out tomorrow, would be to look at any bounds that the moment generating function of $X$ yields. Your integral is a part of the expected value of $\log(1+aX)$, so if you find its moment generating function, you can relate one part of the integral to other parts. 
But again, you'll require strict assumptions on $f(x)$. Even in a Gaussian family, I can shift the mean so far to the left of $b'$ that this integral is as small as desired. Without relationships between the parameters you list, inequality bounds won't be too helpful.
