Question

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: $$f(x) = x - \frac{x^2}{2} - + \frac{2x^3}{3} - \frac{5x^4}{4} + \frac{14x^{5}}{5} - \ldots$$