Fekete-szegö Inequalities Achieved for -Starlike Functions and Their Classical Counterparts

The study of analytic functions is increasingly informed by developments in a mathematical field known as -calculus, which offers new ways to model phenomena in areas like discrete and quantum physics. S. Sivaprasad Kumar and S. Pannu present a systematic investigation into two newly defined subclasses of these functions: -starlike functions linked to the Ma-Minda function, and their classical equivalents. Their research establishes crucial coefficient estimates , including inequalities of Fekete-Szegö, Kruskal, and Zalcman types , providing a deeper understanding of the geometric behaviour of these functions. By also determining bounds for Hankel and Toeplitz determinants, Kumar and Pannu significantly advance the field of geometric function theory and offer tools for future exploration.

Since Jackson’s foundational work on q-differentiation and q-integration, this field has found diverse applications in optimal control theory, fractional calculus, and q-difference equations. The q-derivative operator plays a crucial role in special functions, quantum theory, and statistical mechanics. Scientists pioneered a systematic investigation into two novel subclasses of analytic functions: the q-starlike functions associated with the Ma-Minda function and their classical counterparts.

A methodological framework centered around q-calculus was employed to explore the geometric properties of these function classes, beginning with establishing precise coefficient estimates. The study utilised Fekete-Szegö, Kruskal, and Zalcman-type inequalities to rigorously define the boundaries of these functions, revealing fundamental characteristics of their behaviour. To achieve these bounds, the team developed a series of inequalities, including τ29 −(1 −b2 2) τ26 6q2(1 + q + q2), where τ25 := 12 −12q + 18q2 −5q3, τ27 := 6(1 + 2q2), τ29 := 6q, τ26 := 12 −6q + 18q2 −5q, τ28 := 6(1 −2q + 2q2). These calculations enabled the researchers to determine sharp limits for coefficients like a2, a3, and a4, demonstrating that |a2| ≤1, |a3| ≤1, and |a4| ≤17/18.

Sharpness was confirmed through the use of specifically defined extremal functions, ensuring the accuracy and reliability of the derived bounds. The work further extended to the determination of Hankel and Toeplitz determinants, providing additional insights into the functions’ structural properties. Scientists harnessed analytical techniques to establish that |H2,2(f)| ≤1/4, and derived bounds for Toeplitz determinants, including |T2,3(f)| ≤35/324 and |T3,2(f)| ≤ 1/324. The approach enables a smooth convergence of results as q approaches 1, validating the consistency of the limiting approach and demonstrating the connection between the q-analog and classical function classes. This methodological innovation allows for a unified understanding of both discrete and continuous function behavior.

Ma-Minda Function Coefficient Estimates and Determinants

Scientists have introduced two novel subclasses of analytic functions: the class of -starlike functions associated with the Ma-Minda function ξq, and its classical counterpart associated with ξ, where q is a parameter between 0 and 1. The research meticulously investigates the geometric properties of these function classes, establishing precise coefficient estimates, including Fekete-Szegö, Kruskal, and Zalcman-type inequalities. These inequalities define fundamental limits on the coefficients of the functions, providing a deeper understanding of their behaviour. Experiments revealed bounds for both Hankel and Toeplitz determinants for the newly defined classes of functions.

The team measured the series expansion of ξq(z) as 1 + z + qz + (5/6)qz + (5/6)qz⁴ + (101/120)q⁴z⁵, demonstrating the function’s analytic properties within the open unit disk. Correspondingly, the series expansion for ξ(z) was determined as 1 + z + z + (5/6)z + (5/6)z⁴ + (101/120)z⁵, confirming its classical counterpart’s behaviour. These expansions are crucial for further analysis and application of the functions. Results demonstrate the definition of the q-starlike function class S∗ξq, where zdqf(z)/f(z) is subordinate to ξq(z). Scientists recorded the integral form of these functions, expressed as z exp ∫₀ [ξq(w(t)) − λq]/t dq t, where λq equals ln(q)/(q-1).

The team also derived the explicit series representation of the integral, providing a concrete formula for calculating the functions. The extremal function fq(z) was characterised through a convolution equation: fq(z) ∗ z (1 −qz)(1 −z) = fq(z) · ξq(z), establishing a key relationship for further investigation. Measurements confirm the existence of a unique analytic function, normalised with fq(0) = 0 and f’q(0) = 1, satisfying the functional relation described above. The research introduces the sth Hankel determinant, Hs,n(f), defined by a specific matrix structure involving the function’s coefficients. Establishing sharp upper bounds for these determinants is a central focus, promising advancements in geometric function theory and its applications in areas like quantum physics and statistical mechanics.

Geometric Properties of q-Starlike Functions Established

This study introduced a novel class of q-starlike functions, denoted S∗ξq, defined through the principles of subordination and q-calculus. Researchers systematically investigated the geometric properties of this class, successfully establishing sharp bounds for initial Taylor coefficients, specifically |a2|, |a3|, and |a4|. Several coefficient problems, including Fekete-Szegö, Kruskal, and Zalcman inequalities, were addressed with precise estimates, alongside the derivation of bounds for Hankel and Toeplitz determinants. A significant aspect of this work lies in its ability to unify q-analogue and classical geometric function theory.

As the parameter q approaches 1, the defined class converges to its classical counterpart, S∗ξ, demonstrating a clear relationship between the two frameworks. Importantly, the extremal functions within the classical case are constructed analytically, offering enhanced geometric understanding of the connection between q-deformed and classical theories. The authors acknowledge that the results are limited to the specific function classes investigated, and suggest future work could extend this framework to other q-special functions and explore higher-order coefficient estimations, as well as connections to unresolved problems in geometric function theory. Sivaprasad Kumar and S.

Pannu present a systematic investigation into two newly defined subclasses of these functions: -starlike functions linked to the Ma-Minda function, and their classical equivalents. Their research establishes crucial coefficient estimates, including inequalities of Fekete-Szegö, Kruskal, and Zalcman types, providing a deeper understanding of the geometric behaviour of these functions. By determining bounds for Hankel and Toeplitz determinants, Kumar and Pannu significantly advance the field of geometric function theory and offer tools for future exploration. The theory of q-calculus extends classical analysis by replacing conventional limits with a parameter q, offering a novel approach to mathematical problems.