On several new inequalities (Q2744268)

In the present paper the author has given the following main result.NEWLINENEWLINENEWLINETheorem 1. If \(A_i> 0\), \(\lambda_k> 0\) (\(i= 1,2,\dots, n\); \(k= 1,2,\dots, n\)), \(\sum^n_{i=1} A_i\leq \pi\), \(n\in N_2\), \(0\leq \lambda\leq 1\), then NEWLINE\[NEWLINE{n\choose 2} \Biggl({1-\lambda^2\over 1+\lambda^2}\Biggr) (\lambda\pi)^2\leq(n-1) \sum^n_{k=1} \cos^2 \lambda A_k- 2\cos\lambda\pi \sum_{1\leq i\leq j\leq n}\cos\lambda A_i\cos\lambda A_j\leq {n\choose 2} (\lambda\pi)^2;NEWLINE\]NEWLINE equality holds if and only if \(\lambda= 0\).NEWLINENEWLINENEWLINETwo more results similar to that of Theorem 1 are also given.