A Maclaurin type inequality

The classical Maclaurin inequality asserts that the elementary symmetric means $$s_k(y)=\frac{1}{\left(\genfrac{}{}{0px}{}{n}{k}\right)}\sum _{1\le i_1<\cdots <i_k\le n}{y}_{i_1}\dots y_{i_k}$$ obey the inequality $s_{\mathrm{\ell }}(y{)}^{1\mathrm{/}\mathrm{\ell }}\le s_k(y{)}^{1\mathrm{/}k}$ whenever $1\le k\le \mathrm{\ell }\le n$ and $y=(y_1,\dots ,y_n)$ consists of non-negative reals. We establish a variant $$\mathrm{\mid }{s}_{\mathrm{\ell }}(y){\mathrm{\mid }}^{\frac{1}{\mathrm{\ell }}}\ll \frac{{\mathrm{\ell }}^{1\mathrm{/}2}}{k^{1\mathrm{/}2}}\max (\mathrm{\mid }{s}_k(y){\mathrm{\mid }}^{\frac{1}{k}},\mathrm{\mid }{s}_{k+1}(y){\mathrm{\mid }}^{\frac{1}{k+1}})$$ of this inequality in which the $y_i$ are permitted to be negative. In this regime the inequality is sharp up to constants. Such an inequality was previously known without the $k^{1\mathrm{/}2}$ factor in the denominator.