LAUSR

We present various new inequalities for cosine and sine sums. Among others, we prove that 0.1 $$\begin{aligned} 0\le \sum _{k=0}^n \frac{ (a)_{2k}}{(2k)!} \frac{\cos ((2k+1)x)}{2k+1} \quad {(a\in \mathbb {R})} \end{aligned}$$ 0 ≤ ∑ k = 0 n ( a ) 2 k ( 2 k ) ! cos ( ( 2 k + 1 ) x ) 2 k + 1 ( a ∈ R ) is valid for all $$n\ge 0$$ n ≥ 0 and $$x\in [0,\pi /2]$$ x ∈ [ 0 , π / 2 ] if and only if $$a\in [-2,1]$$ a ∈ [ - 2 , 1 ] , and that 0.2 $$\begin{aligned} 0\le \sum _{k=0}^n \frac{ (b)_{2k}}{(2k)!} \frac{\sin ((2k+1)x)}{2k+1} \quad {(b\in \mathbb {R})} \end{aligned}$$ 0 ≤ ∑ k = 0 n ( b ) 2 k ( 2 k ) ! sin ( ( 2 k + 1 ) x ) 2 k + 1 ( b ∈ R ) holds for all $$n\ge 0$$ n ≥ 0 and $$x\in [0,\pi ]$$ x ∈ [ 0 , π ] if and only if $$b\in [-3,2]$$ b ∈ [ - 3 , 2 ] . Here, $$(a)_n=\prod _{j=0}^{n-1} (a+j)$$ ( a ) n = ∏ j = 0 n - 1 ( a + j ) denotes the Pochhammer symbol. Inequality ( 0.1 ) with $$a=1$$ a = 1 is due to Gasper. We use it to obtain an integral inequality in the complex domain and to provide a one-parameter class of absolutely monotonic functions. An application of ( 0.2 ) leads to a new extension of the classical Fejér–Jackson inequality.