Coefficient estimates and Bohr phenomenon for analytic functions involving semigroup generator

This paper investigates the Bohr phenomenon and sharp coefficient estimates for the class $\mathcal{A}_{\beta}$ of analytic self-maps of the unit disk associated with semigroup generators, establishing improved Bohr and Bohr-Rogosinski radii, solving the Fekete-Szegö problem, and deriving sharp inequalities for logarithmic coefficients and their differences.

The Gist

Imagine you are an architect designing a special kind of building inside a circular room (the "unit disk"). You have a set of blueprints (mathematical functions) that tell you how to stretch and shape this room without tearing it.

This paper is about two main things: how much space these buildings can take up and how predictable their shapes are. The authors, Molla Basir Ahamed and Sanju Mandal, are studying a specific, slightly more complex type of blueprint called $A_\beta$.

Here is a breakdown of their discoveries using simple analogies:

1. The "Bohr Phenomenon": The Safety Radius

Imagine you are walking through a hallway. You know the hallway is safe up to a certain point, but if you go too far, you might bump into a wall.

• The Old Rule: Mathematicians have long known that for a specific type of building, if you stay within a certain distance from the center (about 1/3 of the way out), the sum of all the "stretching instructions" (coefficients) will never exceed the size of the room. This is called the Bohr Radius.
• The New Discovery: The authors asked, "What if we add more rules to our blueprints?" They introduced a new class of buildings ($A_\beta$) that are generated by "semigroups" (think of these as a continuous, smooth flow of water shaping the clay of the building over time).
• The Twist: They didn't just look at the basic stretching; they added "extra ingredients" to the safety check:
  • Multiple Shadows: They looked at how the building casts shadows (Schwarz functions) in different directions.
  • Area Terms: They added a term that accounts for the actual area the building covers, not just the length of the walls.
• The Result: They found a new, precise "Safety Radius" ($R_{\beta}$). As long as you stay inside this radius, the building is guaranteed to fit within the room, even with all these extra complex rules. It's like finding the exact limit of how far you can push a car before it hits a wall, even if the car is carrying extra cargo.

2. The "Fekete-Szegö Problem": Predicting the Shape

Imagine you are trying to guess the shape of a balloon based on how much air is in the first two layers of rubber.

• The Question: If you know the size of the second layer ($a_2$) and the third layer ($a_3$) of your building, can you predict how "bumpy" or "curved" the whole thing will be?
• The Math: There is a famous formula ($a_3 - \mu a_2^2$) that measures this "bumpiness." The value $\mu$ is a dial you can turn to change what you are measuring.
• The Result: The authors calculated the maximum possible bumpiness for their specific class of buildings ($A_\beta$) for every setting of the dial. They found the "sharpest" (most accurate) limits. This is like saying, "No matter how you twist this balloon, it will never get wider than 5 inches."

3. Logarithmic Coefficients: The "Echo" of the Building

Sometimes, instead of looking at the building itself, mathematicians look at the "echo" of the building. This is done using Logarithmic Coefficients.

• The Analogy: Imagine the building has a unique sound. The first note is the "echo" of the entrance, and the second note is the "echo" of the hallway.
• The Question: How different can the volume of the second note be compared to the first note?
• The Result: The authors calculated the exact range of difference between these two "echoes" for both the building and its inverse (if you were to walk backward through the building). They found the tightest possible bounds, meaning they know exactly how much the "echo" can vary.

Why Does This Matter?

In the real world, these mathematical "buildings" model things like fluid flow (how water moves around an object), electrical fields, and even the shapes of galaxies.

• Semigroup Generators: The "flow" mentioned earlier is crucial for understanding how things change continuously over time (like heat spreading or a fluid flowing).
• Sharpness: The authors didn't just give a "good enough" answer; they gave the best possible answer. They proved that you cannot make the safety radius any bigger or the shape predictions any tighter without breaking the rules of the math.

Summary

Think of this paper as a master builder's manual for a specific type of architectural structure.

1. They figured out the exact safe distance you can go before the structure hits the wall, even when you add complex area calculations and multiple shadow effects.
2. They determined the exact limits of how weird the shape can get based on its first few layers.
3. They measured the exact difference between the "echoes" of the structure and its reverse.

They used advanced tools (like hypergeometric functions, which are like super-charged calculators for complex shapes) to prove that their limits are the absolute best possible ones.

Technical Summary

1. Problem Statement

The paper addresses three primary problems within the geometric function theory of analytic functions defined on the unit disk $\mathbb{D}$:

1. Refinement of the Bohr Phenomenon: The classical Bohr inequality states that for bounded analytic functions, the sum of the moduli of the Taylor coefficients converges within a specific radius ($1/3$). The authors seek to improve this radius for the specific class $A_\beta$ by incorporating multiple Schwarz functions and generalized area functionals (specifically the normalized area of the image domain).
2. Sharp Fekete-Szegö Inequality: Determining the sharp upper bound for the functional $|a_3 - \mu a_2^2|$ for functions in the class $A_\beta$, where $\mu$ is a real parameter.
3. Logarithmic Coefficient Differences: Establishing sharp bounds for the moduli differences of logarithmic coefficients ($|\gamma_2| - |\gamma_1|$) and their inverses ($|\Gamma_2| - |\Gamma_1|$) for functions in $A_\beta$ and their inverses.

The Class $A_\beta$:
The study focuses on the subclass $A_\beta \subset \mathcal{A}$ (normalized analytic functions $f(z) = z + \sum_{n=2}^\infty a_n z^n$) defined by the condition:
$$\text{Re}\left( \beta \frac{f(z)}{z} + (1-\beta)f'(z) \right) > 0, \quad 0 \le \beta \le 1.$$
This class is a subclass of $G_0$, the set of holomorphic generators of one-parameter continuous semigroups with the Denjoy-Wolff point at the origin.

2. Methodology

The authors employ a combination of techniques from complex dynamics, geometric function theory, and coefficient estimation:

• Semigroup Theory & Extremal Functions: The authors utilize the structural properties of $A_\beta$ derived from semigroup generators. They rely on the extremal function $\tilde{f}(z)$, defined via the Gauss hypergeometric function ${}_2F_1$, which provides sharp bounds for coefficient estimates and growth theorems in this class.
• Schwarz Functions: The methodology introduces a sequence of distinct Schwarz functions $\{\omega_n(z)\}$ (where $\omega_n(0)=0$ and $|\omega_n(z)|<1$) to replace the standard monomials $z^n$ in the majorant series.
• Area Functional Integration: Instead of limiting refinements to polynomial expressions of the area, the authors introduce a general monotone increasing functional $F(S_r/\pi)$, where $S_r$ is the area of the image of the subdisk $|z|<r$.
• Coefficient Relations: For the Fekete-Szegö and logarithmic coefficient problems, the authors relate the coefficients of $f \in A_\beta$ to the coefficients of Carathéodory functions ($p \in \mathcal{P}$ with $\text{Re } p(z) > 0$) using the relation $\beta \frac{f(z)}{z} + (1-\beta)f'(z) = p(z)$.
• Lemma Application: The proofs rely heavily on:
  • Lemma A: Sharp coefficient estimates $|a_n| \le \frac{2}{n - \beta(n-1)}$.
  • Lemma 3.1: Sharp bounds for $|c_2 - v c_1^2|$ for Carathéodory functions.
  • Lemma 4.1: A specialized lemma for estimating the difference of moduli of linear combinations of coefficients.

3. Key Contributions and Results

A. Refined Bohr and Bohr-Rogosinski Phenomena

The paper generalizes previous results (specifically those by Giri and Kumar) by integrating multiple Schwarz functions and area functionals.

• Theorem 2.1 (Refined Bohr Inequality): For $f \in A_\beta$, the inequality
  $$|\omega_m(z)|^p + \sum_{n=2}^\infty |a_n||\omega_n(z)| + F\left(\frac{S_r}{\pi}\right) \le d(0, \partial f(\mathbb{D}))$$
  holds for $|z| \le R_{\beta, m, p}$. Here, $R_{\beta, m, p}$ is the smallest positive root of a specific equation involving the extremal function $\tilde{f}$ and the functional $F$.

  • Significance: This is the first time the Bohr phenomenon for $A_\beta$ has been established with a general functional $F$ (not just polynomials) and a sequence of distinct Schwarz functions.

• Theorem 2.2 (Refined Bohr-Rogosinski Inequality): A similar sharp radius is derived for the sum involving partial sums of the series and the function value $|f(\omega_m(z))|^p$.

B. Sharp Fekete-Szegö Inequality

The authors solve the classical Fekete-Szegö problem for the class $A_\beta$.

• Theorem 3.1: They provide a piecewise sharp bound for $|a_3 - \mu a_2^2|$ depending on the value of $\mu$ relative to $\beta$:
  • For $\mu < 0$: A specific rational function of $\beta$ and $\mu$.
  • For $0 \le \mu \le \frac{(2-\beta)^2}{3-2\beta}$: The bound is $\frac{2}{3-2\beta}$.
  • For $\mu > \frac{(2-\beta)^2}{3-2\beta}$: Another rational function.
  • Extremal Functions: The bounds are attained by specific rotations of the extremal function $\tilde{f}$ and functions related to $p(z) = \frac{1+z^2}{1-z^2}$.

C. Logarithmic Coefficient Differences

The paper establishes sharp bounds for the differences of the moduli of the first two logarithmic coefficients for both the function and its inverse.

• Theorem 4.1 (Direct Function): For $f \in A_\beta$, the difference $|\gamma_2| - |\gamma_1|$ satisfies:
  $$-\frac{1}{\sqrt{5 - 6\beta + 2\beta^2}} \le |\gamma_2| - |\gamma_1| \le \frac{1}{3-2\beta}$$
• Theorem 4.2 (Inverse Function): For the inverse function $f^{-1}$, the difference $|\Gamma_2| - |\Gamma_1|$ satisfies:
  $$-\frac{1}{\sqrt{3(3-2\beta)}} \le |\Gamma_2| - |\Gamma_1| \le \begin{cases} \frac{1}{3-2\beta} & 0 \le \beta < 1 \\ \frac{-1+5\beta-3\beta^2}{(2-\beta)^2(3-2\beta)} & \beta = 1 \end{cases}$$
  These results generalize known bounds for the class of functions with bounded turning ($\beta=0$).

4. Significance and Impact

• Generalization of Bohr Phenomenon: The paper moves beyond the classical "polynomial" refinement of the Bohr radius. By allowing a general monotone functional $F(S_r/\pi)$ and multiple Schwarz functions, it provides a more robust and flexible framework for studying Bohr-type inequalities in semigroup generator classes.
• Sharpness: All derived radii and coefficient bounds are proven to be sharp, meaning there exist specific extremal functions in $A_\beta$ that attain these bounds. This is crucial for the completeness of geometric function theory results.
• Unification: The results unify and extend previous findings on the Bohr phenomenon, Fekete-Szegö problems, and logarithmic coefficients, specifically bridging the gap between semigroup generators and classical univalent function subclasses.
• Methodological Advancement: The use of the hypergeometric function $\tilde{f}$ as a universal extremal function for the class $A_\beta$ across different problems (Bohr, Fekete-Szegö, Logarithmic) demonstrates a cohesive approach to analyzing this specific subclass.

In summary, this work significantly advances the understanding of the geometric properties of analytic functions generated by semigroups, offering precise, sharp estimates for fundamental problems in the field.