Source-linked AI summary
Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions
Rosihan M. Ali, Lee See Keong, V. Ravichandran, Shamani Supramaniam
TL;DR
The paper studies initial-coefficient estimates for normalized analytic functions whose functions and inverses satisfy Ma-Minda-type subordination conditions. It develops estimates for bi-starlike, bi-convex, and related classes, while connecting the results to earlier work.
Problem
Earlier studies considered bi-univalent subclasses and initial-coefficient estimates, motivating broader Ma-Minda-type coefficient results for functions and their inverses.
Method
The paper uses subordination to a normalized analytic function with positive real part and a range symmetric about the real axis, together with inverse-function expansions and coefficient calculations.
Results
Initial-coefficient estimates are obtained for bi-starlike and bi-convex Ma-Minda-type functions and several related classes.
Takeaways & Limitations
The results extend the study of bi-univalent starlike and convex subclasses and establish connections with earlier known results.
Takeaways & Limitations
Sharp estimates for |a2|, |a3|, and other coefficients remain open, and even non-sharp estimates for |an| with n ≥ 4 would be of interest.
Abstract
from arXiv · showhide
Estimates on the initial coefficients are obtained for normalized analytic functions $f$ in the open unit disk with $f$ and its inverse $g=f^{-1}$ satisfying the conditions that $zf'(z)/f(z)$ and $zg'(z)/g(z)$ are both subordinate to a starlike univalent function whose range is symmetric with respect to the real axis. Several related classes of functions are also considered, and connections to earlier known results are made.
1. Introduction
The paper places bi-univalent starlike and convex function classes within the Ma-Minda subordination framework. It obtains initial-coefficient estimates for these classes and relates them to earlier results.
- Background: Bi-univalent functions require both f and its inverse f^-1 to be univalent in the unit disk.The class is denoted by σ.
- Ma-Minda framework: Subordination unifies subclasses characterized by zf'(z)/f(z) or 1+zf''(z)/f'(z) lying in suitable domains.Ma-Minda functions use a superordinate function with positive real part and a range starlike with respect to 1 and symmetric about the real axis.
- Ma-Minda framework: Bi-starlike and bi-convex Ma-Minda classes impose the corresponding Ma-Minda conditions on both f and f^-1.They are denoted STσ(ϕ) and CVσ(ϕ), respectively.
- Related work: Earlier work studied bi-univalent subclasses analogous to strongly starlike, starlike, and convex functions and derived initial-coefficient estimates.The paper builds on this line of research and on classes investigated in earlier work.
- Contribution: The paper obtains initial-coefficient estimates for bi-starlike and bi-convex Ma-Minda functions, considers related classes, and connects the results to earlier findings.These classes are motivated by corresponding classes investigated in prior work.
2. Coefficient Estimates
The paper derives initial coefficient estimates for several Ma–Minda bi-univalent function classes defined through subordinations involving f, its inverse, and a symmetric starlike function ϕ. Special parameter choices recover earlier results for bi-starlike and bi-convex subclasses, while sharpness remains open.
- ϕ is analytic with positive real part, normalized by ϕ(0)=1 and ϕ′(0)>0, and maps the disk to a real-axis-symmetric region.
- STσ(α,ϕ): Theorem 2.2 establishes estimates for the class STσ(α,ϕ), with the α=0 case yielding Ma–Minda bi-starlike estimates.The proof combines coefficient identities from f, its inverse, and the subordinations before applying positive-real-part inequalities.
- The coefficient estimates are obtained by expressing f and its inverse through subordinations and auxiliary functions with positive real part.The auxiliary functions satisfy coefficient bounds such as |b_i|≤2 and |c_i|≤2, which are used in the estimates.
- Mσ(α,ϕ): Theorem 2.3 treats the bi-Mocanu-convex class Mσ(α,ϕ), which unifies Ma–Minda bi-starlike and bi-convex classes.At α=0 it gives bi-starlike estimates, while α=1 gives the bi-convex estimates stated in Corollary 2.2.
- Lσ(α,ϕ): Theorem 2.4 derives coefficient estimates for Lσ(α,ϕ), another class reducing to the Ma–Minda bi-starlike and bi-convex classes.
- Sharp estimates for |a2| and |a3|, and even non-sharp estimates for |a_n| with n≥4, remain open problems for the investigated classes.