[Paper Review] The titled Carathodory class and its applications

Limei Wang|arXiv (Cornell University)|Mar 9, 2010

Analytic and geometric function theory19 references3 citations

TL;DR

This paper introduces and systematically studies the tilted Carathéodory class $\mathcal{P}_{\lambda}$, defined as analytic functions mapping the unit disk into a right half-plane rotated by angle $\lambda$. It establishes equivalent characterizations using integral representations, subordination, and duality, derives sharp estimates for functionals like coefficients and distortion, and applies these results to characterize $\lambda$-spirallike, close-to-convex, and Robertson functions, with explicit extremal functions provided for key inequalities.

ABSTRACT

This survey mainly deals with the tilted Carath{é}odory class by angle $λ$ (denoted by $\mathcal{P}_λ$) an element of which maps the unit disc into the tilted right half-plane $\{w: \Re e^{iλ} w>0\}$. Firstly we will characterize $\mathcal{P}_λ$ from different aspects. In section 3, various estimates of functionals over $\mathcal{P}_λ$ are deduced from the known corresponding estimates of $\mathcal{P}_{0}$ or elementary functional analysis. Finally some subsets of analytic functions related to $\mathcal{P}_λ$ including close-to-convex functions with argument $λ$, $λ$-spirallike functions, $λ$-Robertson functions and analytic functions whose derivative is in $\mathcal{P}_λ$ are also considered as applications

Motivation & Objective

• To define and characterize the tilted Carathéodory class $\mathcal{P}_{\lambda}$, a generalization of the classical Carathéodory class $\mathcal{P}_0$ to functions mapping into a tilted right half-plane.
• To derive sharp estimates for key functionals—such as the $n$-th coefficient, distortion, and growth—over $\mathcal{P}_{\lambda}$ using extremal functions and functional analysis.
• To apply the theory of $\mathcal{P}_{\lambda}$ to geometric function theory, particularly to classes like $\lambda$-spirallike, close-to-convex with argument $\lambda$, $\lambda$-Robertson, and functions with derivative in $\mathcal{P}_{\lambda}$.

Proposed method

• Characterize $\mathcal{P}_{\lambda}$ via linear transformation: $p \in \mathcal{P}_{\lambda}$ iff $\frac{e^{i\lambda}p - i\sin\lambda}{\cos\lambda} \in \mathcal{P}_0$, linking it to the classical class.
• Use integral representation: $p(z) = \int_{\partial\mathbb{D}} \frac{1 + e^{-2i\lambda}xz}{1 - xz} d\mu(x)$ for a Borel probability measure $\mu$ on the unit circle.
• Establish subordination: $p \prec p_\lambda$, where $p_\lambda(z) = \frac{1 + e^{-2i\lambda}z}{1 - z}$, which maps $\mathbb{D}$ onto the tilted half-plane $\mathbb{H}_\lambda$.
• Define dual and second dual sets: $V^* = \{g \in \mathcal{A}_0 : (f*g)(z) \neq 0 \text{ in } \mathbb{D} \text{ for all } f \in V\}$, and show $\mathcal{P}_\lambda = V_\lambda^*$ for a specific $V_\lambda$.
• Derive sharp estimates for $\operatorname{Re} \frac{zp'(z)}{p(z)}$ using extremal functions from $\mathcal{P}_0$, yielding results equivalent to Ruscheweyh and Singh but with explicit extremal functions.
• Apply the theory to geometric function classes by relating them to $\mathcal{P}_\lambda$ via subordination and convolution, e.g., $f \in \mathcal{CL}(\lambda)$ iff $\frac{zf'}{g} \in \mathcal{P}_\lambda$ for some starlike $g$.

Experimental results

Research questions

• RQ1 What are the equivalent characterizations of the tilted Carathéodory class $\mathcal{P}_{\lambda}$, and how do they relate to the classical $\mathcal{P}_0$ class?
• RQ2 What are the sharp estimates for the $n$-th coefficient, distortion, and growth functionals over $\mathcal{P}_{\lambda}$, and what are the extremal functions that achieve them?
• RQ3 How can the theory of $\mathcal{P}_{\lambda}$ be applied to derive sharp distortion and growth theorems for $\lambda$-spirallike, close-to-convex, and $\lambda$-Robertson functions?
• RQ4 What is the sharp bound for the hyperbolic norm $||f||$ of the Pre-Schwarzian derivative for functions $f$ with $f' \in \mathcal{P}_\lambda$, and which functions achieve it?

Key findings

• The class $\mathcal{P}_{\lambda}$ is characterized by the condition $\frac{e^{i\lambda}p - i\sin\lambda}{\cos\lambda} \in \mathcal{P}_0$, providing a direct link to the classical Carathéodory class.
• Sharp estimates for $\operatorname{Re} \frac{zp'(z)}{p(z)}$ are derived, with extremal functions explicitly given, matching but refining the result of Ruscheweyh and Singh.
• The distortion theorem for $\mathcal{CL}(\lambda)$ is sharpened: $\frac{1}{A(\lambda,r)(1+r)^2} \leq |f'(z)| \leq \frac{A(\lambda,r)}{(1-r)^2}$, where $A(\lambda,r)$ is a symmetric function of $\lambda$ satisfying $\frac{1-r}{1+r} \leq A(\lambda,r) \leq \frac{1+r}{1-r}$.
• For $\mathcal{D}(\lambda) = \{f \in \mathcal{A}_1 : f' \in \mathcal{P}_\lambda\}$, the distortion bound $1/A(\lambda,r) \leq |f'(z)| \leq A(\lambda,r)$ is sharp, with extremal function $f(z) = -(1+e^{-2i\lambda})\log(1-z) - e^{-2i\lambda}z$.
• The hyperbolic norm of the Pre-Schwarzian derivative satisfies $||f|| \leq 2$ for all $f \in \mathcal{D}(\lambda)$, and this bound is sharp, achieved by the same extremal function.
• The extremal functions for all sharp estimates are explicitly constructed, enabling precise distortion and growth theorems for $\lambda$-spirallike and close-to-convex functions with argument $\lambda$.