Geometric Function Theory of Analytic Functions

Geometric Function Theory of Analytic Functions addresses the study of holomorphic maps in the complex plane, emphasising their geometric properties under conformal mapping. Univalent functions on the unit disk—those that are injective and holomorphic—form the backbone of the field, with central problems focusing on coefficient estimates, growth and distortion theorems, boundary regularity and extremal mappings. Seminal achievements, such as the resolution of the Bieberbach conjecture, highlight the deep interplay between algebraic coefficient bounds and geometric shape control. Subclasses defined by starlikeness and convexity capture functions mapping the disk onto star-shaped or convex domains, respectively, prompting detailed investigations of Hankel determinants and Fekete–Szegő functionals. More recently, operator techniques, quantum (q-)calculus and convolution methods have been introduced, forging connections between differential subordination and generalised integral transforms. This synthesis of classical and modern approaches has broadened applications to areas such as fluid dynamics, signal processing and computer vision, where geometric criteria underpin stability and distortion control. The global significance of this theory lies in its unification of complex analysis, algebraic methods and geometric intuition, offering a versatile framework for both pure and applied research.

Research from Nature Portfolio

Recent studies have introduced Horadam polynomial techniques to establish sharp coefficient bounds for bi-univalent classes of analytic functions defined on the unit disk. By employing generating functions of Horadam sequences, precise estimates for the second and third Taylor coefficients were obtained, leading to improved Fekete–Szegő inequalities. The analysis demonstrated practical utility in computer vision, where these polynomial coefficient bounds enhance robustness in texture analysis algorithms.

Research from all publishers

A novel approach using a q-convolution differential operator has blended quantum calculus with classical Sălăgean and Janowski frameworks, yielding new subclasses of analytic functions. This method derives significant coefficient inequalities via differential subordination and addresses q-differential equations of Briot–Bouquet type, with implications for information theory and thermodynamics.

In the classical setting, sharp bounds for the second Hankel determinant of univalent functions have been refined. Estimates cover strongly starlike, parabolic starlike and lemniscate starlike functions, clarifying how subordination to specific analytic functions influences determinant behaviour and identifying extremal mappings under various geometric constraints.

Generalised Sălăgean operators have been used to define hierarchies of univalent functions, establishing inclusion relations, extreme-point characterisations and convolution properties. These operator-based classes facilitate systematic derivation of coefficient bounds and subordination results, enriching the toolkit for geometric classification and offering new avenues for extremal problems.

Geometric Function Theory of Analytic Functions publication trend

The graph below shows the total number of articles in geometric function theory of analytic functions across all publications each year (not limited to Nature Index journals).

Technical terms

Analytic function: A complex function differentiable at every point in its domain.

Univalent function: An injective analytic function, mapping a domain conformally onto its image.

Starlike function: A univalent function whose image is star-shaped with respect to the origin.

Convex function: A univalent function mapping to a convex domain.

Bi-univalent function: An analytic function whose inverse is also univalent in the unit disk.

Hankel determinant: A determinant constructed from successive Taylor coefficients, used to quantify nonlinear interactions among coefficients.

Subordination: A relation where one analytic function is represented as the composition of another with a Schwarz function mapping the disk into itself.

q-derivative: A generalised difference operator from quantum calculus that extends the classical derivative concept.

References

1. Texture analysis using Horadam polynomial coefficient estimate for the class of Sakaguchi kind function. Scientific Reports (2023).
2. Geometric process solving a class of analytic functions using q-convolution differential operator. Journal of Taibah University for Science (2020).
3. Bounds for the second Hankel determinant of certain univalent functions. Journal of Inequalities and Applications (2013).
4. On univalent functions defined by a generalized Sălăgean operator. International Journal of Mathematics and Mathematical Sciences (2004).