# Matrix inequality : trace of exponential of Hermitian matrix

I want to know whether the following inequality holds or not. \begin{align} (\mathrm{Tr}\exp[(A+B)/2])^2\leq(\mathrm{Tr}\exp A)(\mathrm{Tr}\exp B)\tag{1} \end{align} where $$A, B$$ are Hermitian matrices of the dimension $$D$$.

Note that if $$A$$ and $$B$$ commute, we can see (1) holds using the simultaneously diagonalizing basis and Cauchy-Schwarz inequality. The problem is the case where $$A$$ and $$B$$ do not commute.

You can prove it using the Golden-Thompson inequality $$Tr (e^{A+B}) \leqslant Tr(e^{A} e^{B})$$ and then applying the Cauchy-Schwarz inequality.