[Paper Review] Starlike Functions associated with a Petal Shaped Domain

S. Sivaprasad Kumar, Kush Arora|arXiv (Cornell University)|Oct 20, 2020

Analytic and geometric function theory20 references19 citations

TL;DR

This paper introduces a new subclass of starlike functions, denoted $\mathcal{S}^{*}_{\rho}$, defined by subordination to the function $1 + \sinh^{-1}(z)$, which maps the unit disk onto a petal-shaped domain. The authors establish sharp radius results for this class and related function families, including $\mathcal{F}_1$, $\mathcal{F}_2$, and $\mathcal{F}_3$, providing exact values for the largest $r$ such that $f(rz)/r \in \mathcal{S}^{*}_{\rho}$ for $f$ in each class.

ABSTRACT

This paper deals with some radius results and inclusion relations that are established for functions in a newly defined subclass of starlike functions associated with a petal shaped domain.

Motivation & Objective

• To define and study a new subclass of starlike functions associated with a petal-shaped domain in the complex plane.
• To establish sharp radius results for the new class $\mathcal{S}^{*}_{\rho}$ and related function families.
• To explore inclusion relations and geometric properties of functions subordinate to $1 + \sinh^{-1}(z)$.
• To derive exact values for the $\mathcal{S}^{*}_{\rho}$-radius for various function classes, ensuring sharpness through extremal functions.

Proposed method

• Define the class $\mathcal{S}^{*}_{\rho}$ via subordination: $\frac{zf'(z)}{f(z)} \prec 1 + \sinh^{-1}(z)$, where $1 + \sinh^{-1}(z)$ maps $\mathbb{D}$ to a petal-shaped region.
• Utilize the known subordination method and properties of analytic functions in the unit disk to derive bounds on the function $zf'(z)/f(z)$.
• Apply Lemma 3.1 to estimate $\left| \frac{zf'(z)}{f(z)} - 1 \right|$ for functions in $\mathcal{F}_1$, $\mathcal{F}_2$, and $\mathcal{F}_3$, using real parts of ratios of analytic functions.
• Construct extremal functions such as $f_0(z) = z(1+z^n)/(1-z^n)^2$ and $f_0(z) = z(1+z^n)^2/(1-z^n)$ to verify sharpness of the radius estimates.
• Use the power series expansion of $\sinh^{-1}(z)$ to derive the extremal function $f_0(z) = z \exp\left(\int_0^z \frac{\sinh^{-1}(t)}{t} dt\right)$ for $\mathcal{S}^{*}_{\rho}$.
• Verify sharpness by showing equality is achieved at specific points on the boundary of the unit disk, such as $z = R e^{i\pi/n}$.

Experimental results

Research questions

• RQ1 What are the sharp radius values such that $f(rz)/r \in \mathcal{S}^{*}_{\rho}$ for functions $f$ in the class $\mathcal{F}_1$?
• RQ2 How do the radius estimates for $\mathcal{S}^{*}_{\rho}$ vary for the classes $\mathcal{F}_2$ and $\mathcal{F}_3$?
• RQ3 What is the exact $\mathcal{S}^{*}_{\rho}$-radius for the class $\mathcal{BS}^{*}(0)$, and how does it compare to other known classes?
• RQ4 Can the extremal function $f_0(z) = z \exp\left(\int_0^z \frac{\sinh^{-1}(t)}{t} dt\right)$ be used to achieve sharp bounds in the class $\mathcal{S}^{*}_{\rho}$?
• RQ5 Is there a functional relationship between the petal-shaped class $\mathcal{S}^{*}_{\rho}$ and the crescent-shaped class $\Delta^{*}$ via exponential transformation?

Key findings

• The sharp $\mathcal{S}^{*}_{\rho}$-radius for the class $\mathcal{F}_1$ is $R_{\mathcal{S}^{*}_{\rho,n}}(\mathcal{F}_1) = \left(\frac{\sqrt{4n^2 + (\sinh^{-1}(1))^2} - 2n}{\sinh^{-1}(1)}\right)^{1/n}$.
• The sharp $\mathcal{S}^{*}_{\rho}$-radius for $\mathcal{F}_2$ is $R_{\mathcal{S}^{*}_{\rho,n}}(\mathcal{F}_2) = \left(\frac{\sqrt{9n^2 + 4\sinh^{-1}(1)(n + \sinh^{-1}(1))} - 3n}{2(n + \sinh^{-1}(1))}\right)^{1/n}$.
• The $\mathcal{S}^{*}_{\rho}$-radius for $\mathcal{F}_3$ equals that of $\mathcal{F}_2$, with $R_{\mathcal{S}^{*}_{\rho,n}}(\mathcal{F}_3) = R_{\mathcal{S}^{*}_{\rho,n}}(\mathcal{F}_2)$.
• The extremal function $f_0(z) = z \exp\left(\int_0^z \frac{\sinh^{-1}(t)}{t} dt\right)$ yields the sharp bound for $\mathcal{S}^{*}_{\rho}$, with series expansion $f_0(z) = z + z^2 + \frac{z^3}{2} + \frac{z^4}{9} - \frac{z^5}{72} - \cdots$
• The $\mathcal{S}^{*}_{\rho}$-radius for $\mathcal{BS}^{*}(0)$ is $\sinh^{-1}(1) \approx 0.881374$, which is larger than that for $\mathcal{F}_1$.
• The result is sharp, as demonstrated by extremal functions achieving equality in the bound $\left| \frac{zf'(z)}{f(z)} - 1 \right| = \sinh^{-1}(1)$ at $z = R e^{i\pi/n}$.