The [Fejer-Jackson inequality](https://link.springer.com/article/10.1023/A:1006560422934) as follows:
$$\sum\_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$



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> I conjecture that the inequality as follows holds:
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> $$\sum\_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$$
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> How to prove this inequality? Can You give a comment or a proof or a reference?
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> 



$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$
Comment by Cherng-tiao Perng converted to an answer: It appears that Theorem A of [this paper](https://ac.els-cdn.com/S0898122111008054/1-s2.0-S0898122111008054-main.pdf?_tid=21771cc0-73cc-48ba-afa1-3ae25989a942&acdnat=1530100799_f17da2900208a8df38cfe852dc50b0c9) solves your problem. 


