Riv. Mat. Univ. Parma, Vol. 12, No. 2, 2021

Horst Alzer [a] and Man Kam Kwong [b]

Inequalities for sine and cosine polynomials
Pages: 301-317
Received: 13 February 2021
Accepted in revised form: 23 March 2021
Mathematics Subject Classification: 26D05, 26D15, 33B10.
Keywords: Sine polynomials, cosine polynomials, inequalities.
[a]: Morsbacher Strasse 10, 51545 Waldbröl, Germany.
[b]: The Hong Kong Polytechnic University, Hunghom, Hong Kong.

Abstract:
In this paper, we prove that, letting $$\lambda$$ be a real number,

$$(i) \qquad \lambda \, \sum_{k=1}^n (-1)^k \sin(kx) \leq \sum_{k=1}^n \frac{\sin(kx)}{k}$$

is valid for all $$n\geq 1$$ and $$x\in [0,\pi]$$ if and only if $$\lambda \in [0,2]$$. This extends the classical Fejér-Jackson inequality which states that $$(i)$$ holds for $$\lambda=0$$. An application of $$(i)$$ reveals if $$a>0$$ and $$b$$ are real numbers, then

$$(ii) \qquad \frac{41}{96}+ \sum_{k=1}^n \frac{\cos(kx)}{k+1} \geq a \bigl( \cos(x)+b\bigr)^2$$

holds for all $$n\geq 2$$ and $$x\in[0,\pi]$$ if and only if $$a\leq 2/75$$ and $$b=3/8$$. This refines a result of Koumandos (2001) who proved that the expression on the left-hand side of $$(ii)$$ is nonnegative for all $$n\geq 2$$ and $$x\in[0,\pi]$$. The cosine polynomial in $$(ii)$$ was first studied by Rogosinski and Szegö in 1928.

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