Analytic Bounding in Function Theory

• Analytic bounding is a rigorous method for obtaining explicit bounds on coefficients in analytic functions, leveraging concepts such as subordination and convexity.
• It employs extremal functions and optimization techniques to generalize and improve classical coefficient estimates in univalent and close-to-convex function theory.
• The approach unifies various analytic bounding strategies, offering superior coefficient estimates with practical applications in geometric function theory and operator analysis.

Analytic bounding refers to a family of rigorous techniques for deriving explicit, sharp, and often extremal bounds on functionals or coefficients associated with analytic (holomorphic) objects in mathematical analysis, applied mathematics, and mathematical physics. Such bounds are foundational in univalent function theory, complex approximation, operator theory, the study of special function coefficients, and in quantifying continuity properties or the propagation of local information under analytic constraints. Analytic bounding methods often leverage principles such as subordination, convexity, singular value decay in operator frameworks, extremal majorization, as well as affine and convex optimization in the case of function spaces constrained by analytic and positivity criteria.

1. Foundational Principles: Subordination and Class Structure

The starting point for analytic bounding in geometric function theory is often the notion of subordination. For normalized analytic functions in the unit disk,

$$A = \{ f:\mathbb{D} \to \mathbb{C} : f(z) = z + \sum_{n=2}^{\infty} a_n z^n \},$$

subordination $f \prec g$ holds if there exists $h:\mathbb{D}\to\mathbb{D}$, $h(0)=0$, such that $f(z) = g(h(z))$. Analytic bounding exploits this structure to define function classes admitting strong coefficient estimates.

Bulut introduced two comprehensive subclasses by parameterized subordination involving two convex univalent functions $\phi$ and $\psi$ in the Ma–Minda family,

$$\begin{align*} K_{\lambda,\delta}(\phi,\psi):\quad & \frac{f'(z)+(\lambda-\delta)zf''(z)+\lambda\delta z^2 f'''(z)}{g'(z)} \prec \phi(z) \ S_{\lambda,\delta}(\phi,\psi):\quad & \frac{(1-\lambda+\delta)f(z)+(\lambda-\delta)z f'(z)+\lambda\delta z^2 f''(z)}{h(z)} \prec \phi(z), \end{align*}$$

where $g \in K(\psi)$ denotes a convex function in the Ma–Minda sense with generator $\psi$, and $f \prec g$0 denotes a starlike function with respect to $f \prec g$1 (Bulut, 2019).

The parameter choice $f \prec g$2 recovers wide classes, including the well-studied Altıntaş–Kılıç classes and classical convex, starlike, and close-to-convex functions.

2. Main Coefficient Bounds: Theorems and Explicit Formulas

The analytic bounding problem in these settings is to give sharp estimates for $f \prec g$3 in the Taylor expansion of $f \prec g$4. Bulut's general result is:

Theorem 3.1 (Convex-type analytic bound):

If $f \prec g$5, for $f \prec g$6,

$f \prec g$7

where $f \prec g$8 holds for coefficients of convex $f \prec g$9 and $h:\mathbb{D}\to\mathbb{D}$0 from convexity of $h:\mathbb{D}\to\mathbb{D}$1.

Consequently,

$h:\mathbb{D}\to\mathbb{D}$2

The starlike-type analog follows identically with $h:\mathbb{D}\to\mathbb{D}$3 and $h:\mathbb{D}\to\mathbb{D}$4 coefficients. These results generalize a large range of prior analytic bounds and reduce to known estimates under specialization.

The proof combines coefficient extraction via subordination, Rogosinski's inequality on convex functions (ensuring $h:\mathbb{D}\to\mathbb{D}$5 for Taylor expansion of the analytic majorant), and convexity-based extremals for coefficient sharpness.

3. Extremality, Sharpness, and Comparison with Classical Bounds

The analytic bounds above are sharp in the sense that extremal functions attaining equality are explicitly constructed. The extremal map for the coefficient bound arises when the subordination expression $h:\mathbb{D}\to\mathbb{D}$6 is taken as $h:\mathbb{D}\to\mathbb{D}$7 for some $h:\mathbb{D}\to\mathbb{D}$8, and $h:\mathbb{D}\to\mathbb{D}$9 (or $h(0)=0$0) is an extremal function in its Ma–Minda class realizing the upper bound $h(0)=0$1. The associated extremal function can be written via integration of the defining subordination relation.

Explicitly,

$h(0)=0$2

from which $h(0)=0$3 is reconstructed to saturate the bound (Bulut, 2019).

Comparison with Altıntaş–Kılıç: For the Altıntaş–Kılıç case (parameters $h(0)=0$4, $h(0)=0$5), the previous bounds are

$h(0)=0$6

whereas the Bulut bounds yield

$h(0)=0$7

demonstrating a uniform improvement for all $h(0)=0$8.

4. Recovery of Classical and Special Cases

These analytic bounding theorems subsume and recover classical results as immediate corollaries:

• Classical convex/univalent case: $h(0)=0$9, $f(z) = g(h(z))$0, $f(z) = g(h(z))$1 yields $f(z) = g(h(z))$2.
• Reade’s bounds for close-to-starlike functions: $f(z) = g(h(z))$3, $f(z) = g(h(z))$4, $f(z) = g(h(z))$5, $f(z) = g(h(z))$6.
• Libera’s close-to-convex of order $f(z) = g(h(z))$7: $f(z) = g(h(z))$8, $f(z) = g(h(z))$9 analogously, gives corresponding order-dependent bounds.

The methodology thus unifies the analytic bounding of a broad spectrum of subclasses of univalent and related analytic functions.

5. Proof Structure and Underlying Analytic Techniques

The proof flow uses:

• Subordination to embed coefficient estimates into a triangular system.
• Rogosinski’s lemma to bound coefficients of convex majorants.
• Coefficient recursion solved via upper bounds for extremal coefficients $\phi$0 arising from the Ma–Minda convexity/starlike control.
• Extremal attainment established via explicit equality cases in the chain of inequalities.

This systematic connection to underlying structures in function theory enables the approach to generalize previous sharp analytic bounds.

6. Broader Context and Applications

Analytic bounding of coefficients is central in geometric function theory, the design of univalent function subclasses, and the investigation of functionals controlled by univalence, convexity, or growth properties. The extension to parameterized Ma–Minda classes encompasses a wide array of settings in geometric analysis, and the bounding methods are tightly linked to sharp estimates in conformal and quasiconformal mapping, as well as to applications in control of boundary behavior for analytic and univalent functions.

The improvement over prior analytic bounds, such as those of Altıntaş–Kılıç, demonstrates that structured analytic bounding is not only a unifying theoretical device but also yields strictly stronger results for coefficient control in classes of analytic functions (Bulut, 2019).

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References:

A. Bulut, "Comprehensive subclasses of analytic functions and coefficient bounds" (Bulut, 2019). O. Altıntaş, O. Kilic, works referenced within (Bulut, 2019).