Doubt from trignometry


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@Hari_Shankar sir please help

It resembles Fejer's inequality but cant find a Wiki link to it. Give me some time

Sir whenever free kindly share your approach
I am trying but not getting idea

@Sneha_2021 what is the source?

@Hari_Shankar sir's_inequality#:~:text=In%20approximation%20theory%2C%20Jackson's%20inequality,function%20or%20of%20its%20derivatives.

Are you talking of this??

  • Sir its concept out of my mind can you explain it little bit??

No idea some one asked me @Samrat_2020

euclid.ijm.1258131058.pdf (248.8 KB)
I this sir is talking about this

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This is all I found

These all solution dont giving idea about upper bound why only it should be less then {2\sqrt{\pi}}

\displaystyle \sum_{k=1}^n \left(\frac{sin{kx}}{k}\right) is a Fourier series of a sawtooth wave

I tried to find a Fourier series of a sawtooth wave but that doesn't go good
All I get that the amplitude of sawtooth wave here is 1

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I found the function
For 0<x<π

[•] is GIF

This one is provable if you take that sine function to be an like this

I Don't know about it
Plz send me proof of it
It may help me @Samrat_2020

I was talking about that pi -x/2 Part

Saw this on stackexchange

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Thanks for this one