On Starlike Functions Associated with a Bean Shaped Domain

Abstract

In this paper, we introduce and explore a new class of starlike functions denoted by $\mathcal{S}^*_{\mathfrak{B}}$, defined as follows: $$\mathcal{S}^*_{\mathfrak{B}}=\{f\in \mathcal{A}:zf'(z)/f(z)\prec \sqrt{1+\tanh{z}}=:\mathfrak{B}(z)\}.$$ Here, $\mathfrak{B}(z)$ represents a mapping from the unit disk onto a bean-shaped domain. Our study focuses on understanding the characteristic properties of both $\mathfrak{B}(z)$ and the functions in $\mathcal{S}^*_{\mathfrak{B}}$. We derive sharp conditions under which $ψ(p)\prec\sqrt{1+\tanh(z)}$ implies $p(z)\prec ((1+A z)/(1+B z))^γ$, where $ψ(p)$ is defined as: \begin{equation*} (1-α)p(z)+αp^2(z)+β\frac{zp'(z)}{p^k(z)}\quad \text{and}\quad (p(z))^δ+β\frac{zp'(z)}{(p(z))^k}. \end{equation*} Additionally, we establish inclusion relations involving $\mathcal{S}^*_{\mathfrak{B}}$ and derive precise estimates for the sharp radii constants of $\mathcal{S}^*_{\mathfrak{B}}$.

Figures (5)

• Figure 1: Graph indicating the sharpness of various bounds (in context of Lemma \ref{['bds']}) associated with $\mathfrak{B}(\mathbb{D}).$
• Figure 2: Boundary curves of best dominants and subordinants of $\mathfrak{B}(z)$.
• Figure 3: Sigmoid domain $\psi(\mathbb{D}_{r_{SG}})$ is sharply contained in $\mathfrak{B}(\mathbb{D}).$
• Figure 4: (A) Sharpness for $\alpha=0.3$, $\psi_{L,0.3}(\partial\mathbb{D}_{r_{L}(0.3)})\subseteq\mathfrak{B}(\mathbb{D})$ and (B) Sharpness for $\alpha=0$, $\psi_{L,0}(\partial\mathbb{D}_{r_{L}(0)})\subseteq\mathfrak{B}(\mathbb{D})$.
• Figure 5: (A) Sharpness for $\alpha=0.3$, $\psi_{e,0.3}(\partial\mathbb{D}_{r_{e}(0.3)})\subseteq\mathfrak{B}(\mathbb{D})$ and (B) Sharpness for $\alpha=0$, $\psi_{e,0}(\partial\mathbb{D}_{r_{e}(0)})\subseteq\mathfrak{B}(\mathbb{D})$.

Theorems & Definitions (41)

• Definition 1.1
• Theorem 2.1
• proof
• Lemma 2.2
• Remark 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Lemma 2.7
• proof