Uniqueness of solutions of a \(H^\infty\) optimization problem and complex geometric convexity (Q1589331)

Denote by \(H^\infty_N\) the space of vector-valued functions \(f= (f_1,f_2,\dots, f_N)\) defined on the unit circle \(T= \{z\in\mathbb{C}, |z|=1\}\) such that each coordinte \(f_j\) extends to be analytic on the entire unit disc. Let \(\Gamma: T\times \mathbb{C}^N\to \mathbb{R}\) be a continuous and nonnegative function. In this paper is considered the following extremal problem (OPT-problem): Find \(f^*\in H^\infty_N\) such that \[ \inf_{f\in H^\infty_N} \sup_{\theta\in \mathbb{R}} \Gamma(e^{i\theta}, f(e^{i\theta}))= \sup_{\theta\in\mathbb{R}} \Gamma(e^{i\theta}, f^*(e^{i\theta})). \] The OPT-problem is central to frequency domain system design problems and is important to the area of \(H^\infty\)-control. In the particular case when \(\Gamma(e^{i\theta}, z)=|g(e^{i\theta})- z|\), the OPT-problem reduces to the Nehari one. In this paper the author proved a new uniqueness result for a type of convexity which is strictly between geometric convexity and pseudoconvexity.