Application of the extremum principle to investigating certain extremal problems

Abstract:

Denote by C, K, X, respectively, a complex plane, the disc $\{ z \in {\textbf {C:}} \left | z \right | < 1\}$ and any compact Hausdorff space. Denote by P a set of probabilistic measures defined on Borel subsets of the space X. For $\mu \in P$, let $f(z) = \int _X {q(z, t) d\mu } , z \in K$, and ${\mathcal {F}} = \{ f: \mu \in P\}$. Consider a finite sequence of real functions ${F_0}, {F_{1,}} \ldots , {F_m}$ defined in the space ${R^{2n}}$. Let ${\zeta _1}, \ldots , {\zeta _k}$ be fixed points of the disc K and $\eta (f) = [\operatorname {re} {f^{(0)}}(\zeta ), \operatorname {im} {f^{(0)}}({\zeta _1}), \ldots , \operatorname {re} {f^{({n_1})}}({\zeta _1}), \operatorname {im} {f^{({n_1})}}({\zeta _1}); \ldots , \operatorname {re} {f^{(0)}}({\zeta _k}), \operatorname {im} {f^{(0)}}({\zeta _k}), \ldots , \operatorname {re} {f^{({n_k})}}({\zeta _k}), \operatorname {im} {f^{({n_k})}}({\zeta _k})]$, where $f \in {\mathcal {F}}, n = {n_1} + \cdots + {n_k} + k$. Let ${F_j}(f) = {F_j}(\eta (f)), j = 0, 1, \ldots , m$. We consider the following extremal problem. Determine a minimum of the functional ${F_0}(f)$ under the conditions ${F_j}(f) \leqslant 0, j = 1, 2, \ldots , m, f \in {\mathcal {F}}$. We apply the extremum principle to solve this problem. In the linear case this problem was investigated in [11].