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Let $g(x)=a_1\sin x+a_2\sin 2x+\ldots+a_n\sin nx$, where $a_1,a_2,\ldots,a_n\in \mathbb{R}$ such that $|g(x)|<|\sin x|\;\forall x\in\mathbb{R}$. Then, can we prove that $|a_1+2a_2+\ldots+na_n|<1$.

I think yes, because we may apply the triangle inequality repeatedly and the boundedness of $\sin$ function. Any ideas. Thanks beforehand.

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    Just an observation: the condition is equivalent to $|g'(0)| < 1$. – 2017-01-05
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    @MartinR thanks, I agree. But, is it possible to prove without calculus using triangle inequality? – 2017-01-05
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    Another proof here: http://math.stackexchange.com/a/382722/42969. – 2017-01-05

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