Alpha-convex functions in complex Banach spaces

Let $\mathbb{B}$ be the unit ball in a complex Banach space $\mathbb{X}$. We study biholomorphic mappings $F:\mathbb{B}\to \mathbb{X}$ of the form $F(z)=f(z)z$ for $z\in \mathbb{B}$, where $f:\mathbb{B}\to \mathbb{C}$ is a holomorphic function satisfying $f(0)=1$. Our main contributions are as follows: Inclusion Results: We establish inclusion conditions for ${\mathcal{M}}_{\alpha }(\mathbb{B})$, class of $\alpha$-convex functions in $\mathbb{B}$ to belong to the classes of starlike mappings or quasi-convex mappings of type $B$ on $\mathbb{B}$ for specified ranges of $\alpha$. Characterization: We prove that $F(z)=\frac{f(l_u(z))}{l_u(z)}z\in {\mathcal{M}}_{\alpha }(\mathbb{B})$ if and only if $f\in {\mathcal{M}}_{\alpha }(\mathbb{U})$, where $f\in \mathcal{S}(\mathbb{U})$, $u\in \mathbb{X}$ with $\mathrm{\parallel }u\mathrm{\parallel }=1$, and $l_u\in T(u)$. Duality: We present a duality relationship between the classes ${\mathcal{S}}^{\ast }(\mathbb{U})$ and ${\mathcal{M}}_{\alpha }(\mathbb{B})$ for $\alpha \ge 0$. Hankel Determinant Bounds: We determine sharp upper bounds for the second Hankel determinant of mappings in ${\mathcal{M}}_{\alpha }(\mathbb{B})$ when $\alpha \ge 0$. The results established here directly extend the previous results to the broader setting of complex Banach spaces.