<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">B</mi> </math></EquationSource> </InlineEquation> of a complex Banach space <i>X</i>, 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {M}(\mathbb {B})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and for starlike mappings. Furthermore, we establish strict inclusions between the classes of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-elastically starlike mappings for different values of the real parameter <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, and between these and some classical classes of biholomorphic mappings. A central part of the paper is devoted to obtaining sharp coefficient estimates and Fekete-Szegö type inequalities for the class of elastically starlike mappings. We first consider the case of one complex variable, followed by the general setting of complex Banach spaces. Moreover, using the results obtained in the general case, we derive estimates for particular elastically starlike mappings, and for elastically starlike mappings on the unit polydisc <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$U^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>U</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>.</p>

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

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2026年6月29日

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Mihai Aron

摘要

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 $\mathbb {B}$ $B$ 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 $\mathcal {M}(\mathbb {B})$ $M (B)$ and for starlike mappings. Furthermore, we establish strict inclusions between the classes of $\alpha $ $α$ -elastically starlike mappings for different values of the real parameter $\alpha $ $α$ , 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 $U^n$ $U^{n}$ in $\mathbb {C}^n$ $C^{n}$ .