<p>We introduce and study a notion of dually Lorentzian polynomials, and show that if <i>s</i> is non-zero and dually Lorentzian then the operator <Equation ID="Equ10"> <EquationSource Format="TEX">\(s(\partial _{x_1},\ldots ,\partial _{x_n}):\mathbb R[x_1,\ldots ,x_n] \rightarrow \mathbb R[x_1,\ldots ,x_n]\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>s</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>∂</mi> <msub> <mi>x</mi> <mn>1</mn> </msub> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>∂</mi> <msub> <mi>x</mi> <mi>n</mi> </msub> </msub> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mi mathvariant="double-struck">R</mi> <mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> <mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </Equation>preserves (strictly) Lorentzian polynomials. From this we conclude that any theory that admits a mixed Alexandrov-Fenchel inequality also admits a generalized Alexandrov-Fenchel inequality involving dually Lorentzian polynomials. As such we deduce generalized Alexandrov-Fenchel inequalities for mixed discriminants, for integrals of Kähler classes, for mixed volumes, and in the theory of valuations.</p>