I would like to understand strong $\mathbb{C}$-linear convexity, as defined in the paper "Cauchy-type integrals in several complex variables" (a domain $D$ with $C^1$ boundary is strongly $\mathbb{C}$-linearly convex if any defining function $\rho$ admits a constant $C>0$ so that $|\sum_{i=1}^n \rho_{z_i}(w)(w_i-z_i)|\geq C\|w-z\|^2$ when $w\in bD$ and $z\in\bar{D}$). Where was this concept first defined? Are strong $\mathbb{C}$-linearly convex domains clearly $\mathbb{C}$-convex? I am trying to prove that strong $\mathbb{C}$-linearly convex domains are pseudoconvex. I have skimmed through the book "Complex Convexity and Analytic Functionals", but it seems that my question is somewhat unrelated to the topics of that book. Any related hints will be appreciated.


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