On Lemniscate starlikeness of analytic functions and its application to special functions

TL;DR

The paper develops a coefficient-based criterion for lemniscate starlikeness of analytic functions in the unit disk, using Cauchy products and bounds on Pochhammer/Gamma terms. A central theorem provides sufficient conditions ensuring $zf'(z)/f(z) \prec \sqrt{1+cz}$, with explicit coefficient-sum bounds involving the key constant $(21/4)^{n/3}$; this framework is then applied to a wide range of special functions. The authors derive concrete parameter thresholds for exponential, confluent hypergeometric, Bessel, Struve, and error-function variants, often improving existing results in the literature. The work thus strengthens the link between coefficient data and geometric properties, enabling sharper lemniscate starlikeness results across many classical functions and suggesting directions for further generalizations.

Abstract

This paper investigates the lemniscate starlikeness of analytic functions by deriving specific conditions on their power series coefficients. The study utilizes the Cauchy product of power series along with key inequalities involving the Pochhammer symbol and the Gamma function. The derived results are further applied to a number of special functions, providing parameter restrictions under which these functions become lemniscate starlike. The findings extend and refine several earlier results in this domain.

Figures (4)

• Figure 1: Image of $\phi_l(\mathbb{D})$.
• Figure 2: Mapping of $\delta(b)$ for $b>0$.
• Figure 3: Special case of \ref{['thm-GBessel']}
• Figure 4: Best lower bound of $\nu$ such that $\mathcal{CP}_{B}(\nu, z) \in S^{\ast}(q_c)$.

Theorems & Definitions (27)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 3.1
• proof