Fekete and Szegö problem for a subclass of quasi-convex mappings in several complex variables

Abstract

Let {K} be the familiar class of normalized convex functions in the unit disk. Keogh and Merkes proved the well-known result that {\max }_{f\in {K}}\left|a_{\mathrm{3}}-\lambda a_{\mathrm{2}}^{\mathrm{2}}\right|\le \max \{\mathrm{1}/\mathrm{3},\left|\lambda -\mathrm{1}\right|\},\lambda \in ℂ, and the estimate is sharp for each λ. We investigate the corresponding problem for a subclass of quasi-convex mappings of type B defined on the unit ball in a complex Banach space or on the unit polydisk in {ℂ}^n. The proofs of these results use some restrictive assumptions, which in the case of one complex variable are automatically satisfied.

Key words： Fekete-Szegö problem; quasi-convex mappings of type A; quasiconvex mappings of type B; quasi-convex mappings of type C

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