How to show $\log \cosh(\sqrt x)$ is concave?

Question

I know the definition of convex and concave functions and the second order condition to justify convexity (concavity). But still, I do not know how to show $\log \cosh(\sqrt x)$ is concave.

Thanks for your help.

Answer 1

For $x>0$ the second derivative is: $$\frac{\text{sech}^2\left(\sqrt{x}\right)}{4x}-\frac{\tanh \left(\sqrt{x}\right)}{4x^{3/2}}=\frac{1}{4x\cosh(\sqrt{x})} \left(\frac{1}{\cosh(\sqrt{x})}-\frac{\sinh \left(\sqrt{x}\right)}{\sqrt{x}}\right)$$ Hence the second derivative is negative (and our function is concave) as soon as we show that for $t>0$, $$f(t):=\sinh(t)\cosh(t)-t=\frac{\sinh(2t)}{2}-t>0$$ which holds because $f(0)=0$ and $f$ is strictly increasing in $[0,+\infty)$ since $f'(t)=\cosh(2t)-1>0$ for $t>0$.