Fekete-Szegö inequality, coefficient estimates and construction methods for elastically starlike mappings on the unit ball of complex Banach spaces

Abstract This paper is devoted to the study of a particular type of starlike mappings, namely, elastically starlike mappings on the unit ball of complex Banach spaces. We first introduce the concept of elastically starlikeness on the unit ball $$\mathbb {B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> of a complex Banach space X , starting from the notion of elastically starlikeness in one complex variable, recently introduced by Aron [6]. In this more general setting, we present methods for constructing such mappings, using results for the class $$\mathcal {M}(\mathbb {B})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:mi>B</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and for starlike mappings. Furthermore, we establish strict inclusions between the classes of $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -elastically starlike mappings for different values of the real parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , and between these and some classical classes of biholomorphic mappings. A central part of the paper is devoted to obtaining sharp coefficient estimates and Fekete-Szegö type inequalities for the class of elastically starlike mappings. We first consider the case of one complex variable, followed by the general setting of complex Banach spaces. Moreover, using the results obtained in the general case, we derive estimates for particular elastically starlike mappings, and for elastically starlike mappings on the unit polydisc $$U^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>U</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> in $$\mathbb {C}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> .