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The Fekete-Szegö problem for starlike mappings and nonlinear resolvents of the Carathéodory family on the unit balls of complex Banach spaces. (English) Zbl 1477.30018

Summary: In this paper, we first give a coefficient inequality for holomorphic functions on the unit disc \(\mathbb{U}\) in \(\mathbb{C}\) which are subordinate to a holomorphic function \(p\) on \(\mathbb{U}\) with \(p'(0)\ne 0\). Next, as applications of this theorem, we will give the Fekete-Szegö inequality for subclasses of normalized starlike mappings and normalized quasi-convex mappings of type \(B\) on the unit ball \(\mathbb{B}\) of a complex Banach space. We also give the Fekete-Szegö inequality for \((1+r)J_r\), where \(J_r=J_r[f]\) is the nonlinear resolvent of a mapping \(f\) in the Carathéodory family \(\mathscr{M}(\mathbb{B})\). Various particular cases will be also considered.

MSC:

30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
32K12 Holomorphic maps with infinite-dimensional arguments or values
46G20 Infinite-dimensional holomorphy
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