Triangular polynomial Inequality

(Fejer-Jackson-growall inequality) fejer conjecture in 1910, trigonometric function series
\ [\ Frac {\ pi-x} {2} = \ sum _ {k = 1} ^ {\ infty} \ frac {\ sin kx} {k }, \ quad 0 <X \ Leq \ pi \]
All parts and
\ [S_n (x) = \ sum _ {k = 1} ^ {n} \ frac {\ sin kx} {k}> 0, \ quad n = 1, 2, \ ldots, 0 <x <\ pi \]

(Turan, 1952) set $ \ {A_k \} (k =, \ ldots) $ to a positive strictly decreasing series. set $ \ displaystyle S_m = \ sum _ {k = 1} ^ {m} B _k \ geq 0 $. if $ S_m> 0 $, \ [\ sum _ {k = 1} ^ {n} a_kb_k = \ sum _ {k = 1} ^ {n-1} S_k (a_k-a _ {k + 1 }) + a_ns_n> 0. \]

Exploitation
\ Begin {Align *}
\ Sum _ {k = 1} ^ n {\ sin \ left (2k-1 \ right) x} & =\ sum _ {k = 1} ^ n {\ frac {\ cos 2 \ left (k-1 \ right) x-\ cos 2kx} {\ Text {2} \ SiN x }}
\\
& =\ Frac {1-\ cos 2nx} {\ Text {2} \ SiN x }=\ frac {\ sin ^ 2nx} {\ SiN x}
\\
\ Frac {d} {DT} \ left [\ frac {\ sin 2kt} {2 k \ left (\ sin t \ right) ^ {2 k }}\ right] & =- \ frac {\ sin \ left (2k-1 \ right) T }{\ left (\ sin t \ right) ^ {2 k + 1 }},
\ End {Align *}
Yes
\ [\ Frac {\ sin kx} {k} = 2 \ int _ {X/2} ^ {\ PI/2} {\ left (\ frac {\ sin \ frac {x} {2 }}{\ sin \ Theta} \ right) ^ {2 k} \ frac {\ sin \ left (2k-1 \ right) \ Theta} {\ sin \ Theta} d \ Theta}, \]
Therefore
\ [\ Sum _ {k = 1} ^ n {\ frac {\ sin kx} {k }}= 2 \ int _ {X/2} ^ {\ PI/2} {\ left [\ sum _ {k = 1} ^ n {\ left (\ frac {\ sin \ frac {x} {2 }{\ sin \ Theta }\ right) ^ {2 k} \ frac {\ sin \ left (2k-1 \ right) \ Theta} {\ sin \ Theta} \ right] d \ Theta }. \]

Note
\ [R ^ {2 k }=\ left (\ frac {\ sin \ frac {x} {2 }}{\ sin \ Theta} \ right) ^ {2 k}, \ quad k = 1, 2, \ ldots \]
Obviously $ R ^ {2 k} $ is decreasing, when $0 <x <\ Pi, \ frac {x} {2} \ Leq \ Theta \ Leq \ frac {\ PI} {2} $
\ [\ Sum _ {k = 1} ^ n {R ^ {2 k} \ sin \ left (2k-1 \ right) \ Theta}> 0, \ qquad 0 <\ Theta <\ Pi, \]
This evidence is obtained.

Triangular polynomial Inequality