[Paper Review] Certain classes of bi-univalent functions related to Shell-like curves connected with Fibonacci numbers

N. Magesh, V. K. Balaji+1 more|arXiv (Cornell University)|Oct 15, 2018

Analytic and geometric function theory18 references4 citations

TL;DR

This paper introduces new classes of bi-univalent functions associated with shell-like curves and Fibonacci numbers, deriving estimates for the second and third Taylor-Maclaurin coefficients and Fekete-Szegö inequalities. Using subordination and coefficient bounds via a generalized differential operator, it establishes sharp bounds involving the golden ratio-related parameter $\tau = (1 - \sqrt{5})/2$, extending prior results on bi-univalent functions in the complex plane.

ABSTRACT

Recently, in their pioneering work on the subject of bi-univalent functions, Srivastava et al. \cite{HMS-AKM-PG} actually revived the study of the coefficient problems involving bi-univalent functions. Inspired by the pioneering work of Srivastava et al. \cite{HMS-AKM-PG}, there has been triggering interest to study the coefficient problems for the different subclasses of bi-univalent functions. Motivated largely by Ali et al. \cite{Ali-Ravi-Ma-Mina-class}, Srivastava et al. \cite{HMS-AKM-PG} and Güney et al. \cite{HOG-GMS-JS-Fib-2018} in this paper, we consider certain classes of bi-univalent functions related to shell-like curves connected with Fibonacci numbers to obtain the estimates of second, third Taylor-Maclaurin coefficients and Fekete - Szegö inequalities. Further, certain special cases are also indicated. Some interesting remarks of the results presented here are also discussed.

Motivation & Objective

• To introduce and study new subclasses of bi-univalent functions related to shell-like curves and Fibonacci numbers.
• To derive sharp estimates for the second and third Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ in these classes.
• To establish Fekete-Szegö type inequalities for the coefficient functional $|a_3 - \mu a_2^2|$.
• To generalize and extend previous results on coefficient bounds for bi-univalent functions in the unit disk.

Proposed method

• The authors define new classes $\mathcal{F}\mathcal{S}\mathcal{L}_\Sigma(\gamma,\lambda,\tilde{p})$, $\mathcal{B}\mathcal{S}\mathcal{L}_\Sigma(\gamma,\alpha,\tilde{p})$, and $\mathcal{H}\mathcal{S}\mathcal{L}_\Sigma(\gamma,\tilde{p})$ using subordination with a generating function $\tilde{p}(z)$ linked to Fibonacci numbers and the golden ratio.
• They employ the differential operator $\mathcal{D}^n$ to define the analytic functions in the unit disk $\mathbb{D}$, ensuring univalence and bi-univalence.
• The key technique involves expressing coefficient bounds via subordination and applying the Carathéodory function class $\mathcal{P}$ to control real parts of analytic functions.
• The Fekete-Szegö inequality is derived using a functional $h(\mu)$ that depends on the parameter $\mu$, with piecewise bounds based on the magnitude of $|h(\mu)|$.
• The parameter $\tau = (1 - \sqrt{5})/2 \approx -0.618$ is central, arising from the golden ratio and defining the shell-like curve's geometry.
• The results are validated by showing that special cases reduce to known bounds in [10] and [23], confirming consistency with prior work.

Experimental results

Research questions

• RQ1 What are the sharp estimates for the second and third Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ in bi-univalent functions related to shell-like curves and Fibonacci numbers?
• RQ2 How do Fekete-Szegö inequalities behave for the coefficient functional $|a_3 - \mu a_2^2|$ in these new classes?
• RQ3 What role does the parameter $\tau = (1 - \sqrt{5})/2$ play in shaping the coefficient bounds and geometric properties of the functions?
• RQ4 How do the new classes generalize or extend previous results on bi-univalent functions, particularly those in [10] and [23]?
• RQ5 What are the limiting behaviors of the coefficient bounds when $\gamma$, $\lambda$, or $\alpha$ approach zero or other critical values?

Key findings

• The bound for $|a_2|$ in $\mathcal{F}\mathcal{S}\mathcal{L}_\Sigma(\gamma,\lambda,\tilde{p})$ is $\frac{|\gamma||\tau|}{\sqrt{3\gamma\tau(1+2\lambda) + 4(1-3\tau)(1+\lambda)^2}}$, explicitly depending on $\gamma$, $\lambda$, and $\tau$.
• The bound for $|a_3|$ in the same class is $\frac{4|\gamma||\tau|(1-3\tau)(1+\lambda)^2}{3(1+2\lambda)[3\gamma\tau(1+2\lambda) + 4(1-3\tau)(1+\lambda)^2]}$, showing a nonlinear dependence on $\gamma$, $\lambda$, and $\tau$.
• For the Fekete-Szegö functional $|a_3 - \mu a_2^2|$, the bound is piecewise: $\frac{|\gamma||\tau|}{3+6\lambda}$ if $|h(\mu)| \leq \frac{|\gamma||\tau|}{12+24\lambda}$, and $4|h(\mu)|$ otherwise, with $h(\mu)$ defined as a rational function of $\mu$, $\gamma$, and $\tau$.
• In the special case $\gamma = 1$, $\lambda = 0$, the bounds reduce to those in [10, Corollary 1], confirming consistency with known results for $\mathcal{S}\mathcal{L}_\Sigma(\tilde{p})$.
• For $\mathcal{K}\mathcal{S}\mathcal{L}_\Sigma(\tilde{p})$, the bound for $|a_2|$ is $\frac{|\tau|}{\sqrt{4 - 10\tau}}$, and for $|a_3|$ it is $\frac{|\tau|(1 - 4\tau)}{6 - 15\tau}$, matching [10, Corollary 5].
• The parameter $\tau = (1 - \sqrt{5})/2$ is shown to be fundamental, as it arises from the golden ratio and governs the geometry of the shell-like curve, linking the results to Fibonacci sequences.