Separation principles and Riemann-Hilbert problems
Let \(w_\pm\) be the holomorphic functions on the unit disk \(D\subset \mathbb{C}\) and let \(M_t=\{x+if(t,x)\},\;x\in R,\;t\in\partial D\) where \(f(t,x)\) is real-valued. For a given subset \(A\) in the Hardy space \(H^1\) denote by \(A_+=\{w_+\in A, v_+(t)\geq f(t,u_+(t))\;a.e.\;on\;\partial D\}\) and \(A_-=\{w_-\in A, v_-(t)\geq f(t,u_-(t))\;a.e.\;on\;\partial D\}\) the sets of upper \(w_+=u_++iv_+\) and lower \(w_-=u_-+iv_-\) functions. The author compares the values of upper and lower functions in inner points of \(D\). In particular are studied the conditions on the class \(A\) and separating curve \(M_t\) which guarantee that the values \(w_+ (0)\) of upper functions lie ''above'' the values \(w_- (0)\) of lower functions.
scientific article; zbMATH DE number 1995500
scientific article