Fix a fixed $0<\lambda<0.5$, consider the function \begin{align} f(p) &= \frac{1}{\sqrt{\frac{1 -\lambda + \lambda (1+p)^2}{(1 + \lambda( 1+2p))^2} + \frac{1 -\lambda + 4\lambda (1+p)^2}{(1 + \lambda( 1+2p))^2}-1}}. \end{align} How can I show that $f$ is concave in $p$ in an interval around zero?
I have checked numerically that this is true, and if it helps, we can consider the interval $-0.25<p<0.25$.
By looking at this function, can someone identify a procedure to show concavity, i.e., the steps required? I feel like brute-force differentiation is not the way to go, and it quickly becomes ugly.