Polynomials and pseudoconvexity for Riemann domains over ℂⁿ

We prove that a Riemann domain (G, π) over ℂⁿ is pseudoconvex if and only if for any continuous mapping φ : D̅ ×[0, δ] →G ̿ of the form (π ̿ ◦φ)ⱼ(w, t)=pⱼ(w)+aⱼt (j=1,2, . . . , n), where (G ̿, π ̿) is abstract closure of (G, π), D={w∈ℂ ; |w| < ε}, ε > 0, δ > 0, aⱼ ∈ℂ and pⱼ(w) is a polynomial of w of degree at most 2, with φ(D̅ × (0, δ]) ∪ φ(∂D × {0}) ⊂ G, it follows that φ(D̅ × [0, δ]) ⊂ G.