Starlike, convex, close-to-convex, spiral-like, and $\Phi$-like maps in a commutative Banach algebra with identity

Abstract:

Let $C(X)$ be the space of continuous functions on a compact ${T_2}$-space $X$ where each point of $X$ is a ${G_\delta }$. If $F:B \to C(X)$ is a biholomorphic (in the sense that $F$ and ${F^{ - 1}}$ are Fréchet differentiable) map of $B = \{ f\left | {\left \| f \right \|} \right . < 1\}$ onto a convex domain with $DF(0) = I$, then $F$ is Lorch analytic (i.e., $DF(f)(g) = {a_f}g$ for some ${a_{f}} \in C(X))$). Let $R$ be a commutative Banach algebra with identity such that the Gelfand homomorphism of $R$ into $C(\mathcal {M})$ is an isometry. Starlike, convex, close-to-convex, spirallike and $\Phi$-like functions are defined in $B = \{ x \in R\left | {\left \| x \right \|} \right . < 1\}$ for $L$-analytic functions in $B$ and they are related to associated complex-valued holomorphic functions in $\Delta = \{ z \in \left . \mathbf {C} \right |\left | z \right | < 1\}$.