## Lower bound over a concave function

Hi everyone, I will be to grateful if help me find a tight lower bound $g(x)$ over the following concave function: $$f(x) = \sqrt{1+4x} -1 + \log(\sqrt{1+4x}-1) - \log(2x) \geq g(x),$$ where $x \geq 0$. The taylor expansion around the point ($x = 0$) of this function is given by: $$f(x) = x - \frac{x^2}{2} - + \frac{2x^3}{3} - \frac{5x^4}{4} + \frac{14x^{5}}{5} - \ldots$$

-
 What do require exactly on the function $g$? – Pietro Majer May 30 2011 at 17:59 To be quite tight, and preferably $g(x)=ax+b$ where $b \geq 1$. – Farzad May 30 2011 at 18:37