This topic is not difficult. If we only need to prove uniform boundedness, then we can prove item-by-item integration. Finally, it is proved by DIrchlet integral. This is in Zhou Minqiang's Fractional Exercise.

In fact, we can prove that this upper bound does not exceed 2sqrtπ. This conclusion can be found in the section of trigonometric series in Thomas Thompson's Theory of Functions of Real Variables. The general process is as follows.