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 classesS ₁(D) andS ₂(D) which is similar to the study of the functions of the corresponding classesS ₁(C) andS ₂(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.