Open AccessA SHORT NOTE ON TWO INEQUALITIES FOR SINE POLYNOMIALS
Author(s) -
Horst Alzer
Publication year - 1992
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.23.1992.4538
Subject(s) - mathematics , mathematical proof , sine , combinatorics , inequality , pure mathematics , algebra over a field , discrete mathematics , mathematical analysis , geometry
We present elementary proofs for
\[\sum_{\nu=1}^n(n+1-\nu)\sin(\nu x)>0\]
due to Lukács, and for
\[\sum_{\nu=1}^n\sin(\nu x)+\frac{1}{2}\sin((n+1)x) \quad\quad (*) \]
due to Fejér. Both inequalities are valid for $x \in (0, \pi )$ and $n = 1, 2, \cdots$. Furthermore we determine all cases of equality in (*).
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