Carathéodory Function in Complex Analysis

• Carathéodory functions are analytic functions with strictly positive real parts on domains like the unit disk and the upper half-plane, forming a key concept in complex analysis.
• They admit Herglotz integral representations that link function positivity with probability measures and moment functionals, crucial for spectral theory.
• They are widely applied in interpolation, system theory, and operator theory, influencing studies in boundary behavior and differential subordination.

The Carathéodory function is a central object in complex analysis, operator theory, interpolation, and system theory, defined as an analytic function on specific domains (unit disk or upper half-plane) whose real part is strictly positive. Carathéodory functions admit integral representations (Herglotz formula) that encode their positivity via associated measures, establish deep connections to spectral theory, and serve as the analytic framework for interpolation and extremal problems. In contemporary research, Carathéodory functions appear in boundary interpolation, strongly sectorial conditions, differential subordinations, convex parameterizations, Riemann surface generalizations, and linear spectral transformations.

1. Analytic Definition and Core Properties

A Carathéodory function is any analytic function $f$ on a region $U$ such that $\Re f(z)>0$ for all $z\in U$. On the unit disk $\mathbb{D} = \{z : |z| < 1\}$, this class, typically normalized by $f(0)=1$, is denoted

$$\mathcal{P}_0 = \left\{ p : \mathbb{D} \to \mathbb{C},\ p(0)=1,\ \Re p(z)>0 \right\}.$$

The disk class $P$ is in bijection with the Schur class $$S = \{ w \in \text{Hol}(\mathbb{D}) : |w(z)| \le 1 \}$$ via the Cayley transform:

$$g(z) = \frac{1 + w(z)}{1 - w(z)},\quad w(z) = \frac{g(z) - 1}{g(z) + 1}.$$

On the upper half-plane $U = \{ z \in \mathbb{C} : \Im z > 0 \}$, the Carathéodory class consists of functions $f$ analytic on $U$ satisfying $\Re f(z)>0$, paralleling the Pick class $$P = \{ f \text{ analytic on } U : \Im f(z) \ge 0 \}$$ under different integral representations (Li et al., 2019, Agler et al., 2011, Alpay et al., 2019).

2. Herglotz Integral Representation and Moment Theory

A fundamental property is the Herglotz representation theorem. For $p \in \mathcal{P}_0$, there exists a unique probability measure $\mu$ on $\partial \mathbb{D}$ such that

$$p(z) = \int_{|\zeta|=1} \frac{1 + \zeta z}{1 - \zeta z}\ d\mu(\zeta).$$

In the upper half-plane, the analogous result is

$$f(z) = a + b z + \int_{\mathbb{R}} \left( \frac{1}{t-z} - \frac{t}{1 + t^2} \right) d\mu(t), \qquad \int_{\mathbb{R}} (1 + t^2)^{-1} d\mu(t) < \infty,$$

for real $a \in \mathbb{R}$, $b \geq 0$, and positive finite Borel measures $\mu$ (Alpay et al., 2019, Agler et al., 2011).

This representation encodes the positivity and plays a key role in spectral analysis, mapping the function to moment functionals and measures. In spectral theory and orthogonal polynomials, the Carathéodory formal series naturally encodes Hermitian moment functionals $u$ via

$F(z) = H_0 + 2 \sum_{n=1}^{\infty} u_n z^n,$

where $u_n = \int e^{-i n \theta} d\mu(\theta)$ for $\mu$ supported on $[0, 2\pi)$ (Cantero et al., 2013).

3. Boundary Carathéodory-Fejér Interpolation

The boundary Carathéodory-Fejér problem asks for the construction of $f \in P$ realizing prescribed pseudo-Taylor coefficients at a real boundary point $x \in \mathbb{R}$, that is,

$$\lim_{z \to x} \frac{f^{(k)}(z)}{k!} = a^k, \quad k = 0, \ldots, n,$$

or equivalently, admits a pseudo-Taylor expansion of form $$f(z) = c^0 + c^1 (z-x) + \cdots + c^n (z-x)^n + o((z-x)^n)$$ with coefficients matching $a^k$. The solvability condition is the positivity (or minimal positivity) of the Hankel matrix

$$H_m(a) = [a_{i + j - 1}]_{i, j = 1, \ldots, m}, \quad a_k = \text{data}, \quad n = 2m-1.$$

Julia-Nevanlinna reduction is used inductively to relate higher-order moments to smaller Hankel matrices (Agler et al., 2011).

4. Schur Parameterization and Coefficient Bodies

Any Carathéodory function $g(z) = 1 + \sum_{n=1}^{\infty} a_n z^n$ can be parametrized via the Schur vector $(\gamma_1, \ldots, \gamma_n)$ with modulus $|\gamma_j| \le 1$ as

$a_n = 2 T_n(\gamma_1, \dots, \gamma_n)$

with explicit recurrence relations for $T_n$ in terms of Schur parameters, convexity of the coefficient body, and real-analytic diffeomorphism $D^n \to \text{Int} V_n$, the region of admissible coefficients. Extremal Carathéodory functions arise for modulus one Schur parameters (Li et al., 2019).

5. Strongly Sectorial Carathéodory Functions

Strongly Carathéodory functions of order $\mu$ satisfy sectorial conditions $|\arg p(z)| < \frac{\pi \mu}{2}$ in $U$. Sufficient conditions for membership are derived from subordination principles, notably using Miller–Mocanu lemmas and Carathéodory kernels. Theorems provide explicit inequalities involving real and imaginary parts, or modulus of differential operators, guaranteeing sectorial bounds and relating strongly Carathéodory functions to strongly starlike functions (Shiraishi et al., 2013).

6. Tilted Carathéodory Class and Applications

The tilted Carathéodory class $\mathcal{P}_\lambda$ maps the disk into $\{ w : \Re(e^{i\lambda} w) > 0 \}$ for $| \lambda | < \pi/2$, with corresponding Herglotz integral

$$p(z) = \int_{|\zeta|=1} \frac{1 + e^{-2i\lambda} \zeta z}{1 - \zeta z} d\mu(\zeta).$$

Sharp coefficient bounds $|p_n| \le 2 \cos \lambda$ and distortion theorems are established, and the extremal points are characterized. Functional subclasses, including $\lambda$-spirallike and $\lambda$-Robertson functions, inherit distortion, growth, and norm bounds from $\mathcal{P}_\lambda$ (Wang, 2010, Sharma et al., 2021).

7. Differential Subordination and Geometric Implications

First-order differential subordinations link Carathéodory functions to geometric subclasses (e.g., starlike, close-to-convex). Results give sharp thresholds on parameters (e.g., $\beta$ in $1 + \beta z p'(z) \prec Q(z)$) for dominance by classical Carathéodory functions, utilizing Miller–Mocanu lemmas and boundary value analysis. Sufficient conditions are thus provided for normalized analytic functions to belong to various starlike or sectorial classes (Sharma et al., 2021, Shiraishi et al., 2013).

8. Generalizations to Riemann Surfaces and Operator Theory

On compact real Riemann surfaces $X$ of genus $g > 0$, an additive Carathéodory function is analytic on $X \setminus X_\mathbb{R}$, with $\Re \phi(p) > 0$ for $p$ in the “positive half,” satisfying symmetry under antiholomorphic involutions, and admitting a general Herglotz representation involving the prime form and homology indices:

$$\phi(p) = \int_{X_\mathbb{R}} \left(-\partial_{n_x} \ln E(p,x) + 2\pi i \sum_{k=1}^{g} \omega_k(x) n_k(p) \right) d\nu(x) + i M.$$

de Branges–Rovnyak spaces can be constructed from these functions via Hilbert spaces of bundle sections with explicit kernel formulas (Alpay et al., 2019).

9. Linear Spectral Transformations and Rational Modifications

Given Carathéodory functions $F, G$ related by Laurent polynomials $L(z), M(z)$ and a polynomial $C(z)$ through

$F(z) L(z) = G(z) M(z) + C(z),$

the corresponding moment functionals and measures are connected by algebraic identities and absolute continuity relationships. Rational modifications correspond to the case $C(z) = 0$ and can be characterized by Hermitian symmetry conditions on $(L, M, C)$. Beyond rational modifications, general LST can induce mixing of functionals, support extension, and nontrivial Lebesgue mass terms (Cantero et al., 2013).

---

References:

• (Agler et al., 2011): Pseudo-Taylor expansions and the Carathéodory-Fejér problem
• (Shiraishi et al., 2013): Sufficient conditions for strongly Carathéodory functions
• (Li et al., 2019): Schur parameters and Carathéodory class
• (Alpay et al., 2019): Carathéodory functions on Riemann surfaces and reproducing kernel spaces
• (Wang, 2010): The tilted Carathodory class and its applications
• (Sharma et al., 2021): Differential Subordination implications for Certain Carathéodory functions
• (Cantero et al., 2013): Linear spectral transformations of Carathéodory functions