For sectorial matrices X and Y, the generalized numerical radius satisfies omega_N(XY) <= sec(alpha1)sec(alpha2)omega_N(X)omega_N(Y) and the same bound holds for the Hadamard product.
Alakhrass, Numerical radius of products of special matrices , Journal of Mathematical Inequalities, 17, 997-1006, (2023)
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General Numerical Radius for Products of Sectorial Matrices
For sectorial matrices X and Y, the generalized numerical radius satisfies omega_N(XY) <= sec(alpha1)sec(alpha2)omega_N(X)omega_N(Y) and the same bound holds for the Hadamard product.