A general form of Newton-Maclaurin type inequalities

In this paper, we extend the classical Newton-Maclaurin inequalities to functions $S_{k;s}(x)=E_k(x)+{\text{dsum}}_{i=1}^s{\text{al}}_i{E}_{k-i}(x)$, which are formed by linear combinations of multiple basic symmetric mean. We proved that when the coefficients ${\text{al}}_1,{\text{al}}_2,\cdots ,{\text{al}}_s$ satisfy the condition that the polynomial $$t^s+{\text{al}}_1{t}^{s-1}+{\text{al}}_2{t}^{s-2}+\cdots +{\text{al}}_s$$ has only real roots, the Newton-Maclaurin type inequalities hold for $S_{k;s}(x)$.