On Some Discrete Quadratic Inequalities

摘要: Inequalities of the form $$\alpha \sum\limits_{{\text{j = 0}}}^{\text{n}} {{\text{|}}{{\text{X}}_{\text{j}}}{{\text{|}}^{\text{2}}} \leqslant \sum {{\text{|}}{{\text{X}}_{\text{j}}} \pm {{\text{X}}_{\text{j}}}{\text{ + k}}{{\text{|}}^{\text{2}}} \beta {{\text{|}}{{\text{X}}_{\text{j}}}{{\text{|}}^{\text{2}}}} } }$$ are studied, where x0, …, xn are real or complex variables, α, β constants, 1 ≤ k n, summation (with respect to j) in middle term can be understood four different ways (see introduction) and either plus minus sign is taken. The best constants found all cases. This based on determination eigen-values suitable Hermitean matrices.