The inequality holds trivially at x = 0.

Using Taylor series expansion, $$2sin x = 2(x - rac{x^3}{3!} + O(x^5))$$ and $$ an x = x + rac{x^3}{3} + O(x^5)$$.

Therefore, $$2sin x + an x = 3x - rac{x^3}{3} + rac{x^3}{3} + O(x^5) = 3x + O(x^5)$$.

Since the higher-order terms are non-negative for $$0 le x < {pi over 2}$$, the inequality holds.

The provided argument is incorrect or incomplete and further analysis is needed.