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Suggestion: We have the series expansion $$ \arctan(\pi/z)/\pi=\sum_{k=0}^\infty (-1)^k \frac{\pi^{2k}}{(2k+1)z^{2k+1}} $$ with $z$ respectively $\log(x)$, $\log(y)$, and $\log(x+y)$. This is an alternating series with terms tending to $0$. Since the partial sums alternately overestimates and underestimates the sum, it would suffice to prove the simpler inequality $$ \frac{\pi^2}{3\log(x+y)^3}\le\frac{\pi^2}{3\log(x)^3}+\frac{\pi^2}{3\log(y)^3}-\frac{\pi^4}{5\log(x)^5}-\frac{\pi^4}{5\log(y)^5}. $$