Symmetrization of a Cauchy-like kernel on curves

Given a curve Γ⊂C with specified regularity, we investigate boundedness and positivity for certain three-point symmetrization of Cauchy-like kernel KΓ whose definition is dictated by the geometry complex function theory domains bounded Γ. Our results show that S[ReKΓ] S[ImKΓ] (namely, symmetrizations real imaginary parts KΓ) behave very differently from their counterparts Cauchy previously studied in literature. For instance, quantities S[ReKΓ](z) S[ImKΓ](z) can like 32c2(z) −12c2(z), where z any three-tuple points Γ c(z) Menger curvature z. original kernel, an iconic result M. Melnikov gives symmetrized forms are each equal to 12c2(z) all three-tuples C.