Discrete Wirtinger type inequality under non-periodic end conditions

In this paper, we establish the following two Alzer's type inequalities under non-periodic end conditions, which are discrete analogues of Wirtinger's inequality: Let Z={z1,z2,⋯,zn}(n≥2) be one complex n-tuple with ∑k=1nzk=0, ‖Z‖=∑k=1n|zk|2, ‖ΔZ‖=∑k=1n−1|zk+1−zk|2, min⁡|ΔZ|=min⁡{|z2−z1|,|z3−z2|,⋯,|zn−zn−1|}. Then‖ΔZ‖≥α(n)⋅max1≤k≤n⁡|zk|,‖Z‖≥β(n)⋅min⁡|ΔZ|, where α(n)=6n(n−1)(2n−1) and β(n)=1n⌊n24⌋. The constants α(n) and β(n) are best possible.