Non-linear extremal problems for analytic two-dimensional Riemann-Stieltjes integrals

Abstract

We examine certain non-linear extremal problems for two-dimensional Riemann-Stieltjes integrals\(\varphi (z) \equiv \int {_D \int {g(z,\zeta )d\mu (\zeta ),} \zeta \in D \equiv [\zeta |\left| \zeta \right|} \leqslant 1]\),z∈Δ≡[z‖|z|<1] whereg(z, ζ) is a continuous function in (z, ζ)∈[Δ×D] and an analytic function forz∈Δ and μ(ζ) is a unit mass measure onD. In particular, if the mass is distributed on the segment [a, b], we obtain the well-known Ruscheweyh results for the one-dimensional Riemann-Stieltjes integrals\(\varphi (z) \equiv \int_a^b {g(z,t)d\mu (t),z \in \Delta } \). In particular, ifg(z,σ)≡z/(1—zσ), we determine the maximal domain of univalence and the radii of starlikeness and convexity of order α, −∞<α<1, of the corresponding functions ϕ(z). A particular study is made of the functions of classesS1(D) andS2(D) which is similar to the study of the functions of the corresponding classesS1(C) andS2(C) of Schwarz analytic functions. In addition to obtaining maximal domains of univalence, we also determine the unique extremal functions for each of the functional studied.