Briot-Bouquet differential subordinations of analytic functions involving the Mittag-Leffler function defined in Cardioid domain

TL;DR

This work introduces a new subclass of analytic functions mapped onto a cardioid domain via the Mittag-Leffler operator $\mathcal{E}^{\gamma}_{\alpha,\beta}$. It applies Briot-Bouquet differential subordinations, in the sense of Miller and Mocanu, to derive sharp subordinate relations for this operator with the cardioid target $h_c(z)=1+\frac{4z}{3}+\frac{2z^2}{3}$. Key contributions include explicit best-dominant solutions $q(z)$ (and variants $q_1,q_2$) to the associated Briot–Bouquet equation that govern subordinations for the class $\mathcal{S}^{\gamma}_{\alpha,\beta}(h_c)$, along with extensions under Bernardi–Libera–Livingston perturbations. These results connect Mittag-Leffler operators to cardioid-bound Ma-Minda starlike classes and clarify how subordination is preserved under standard operator transformations, enriching the toolkit of geometric function theory.

Abstract

In this researh work, we establish a new subclass of analytic functions constructed by the Mittag-Leffler function that maps the open unit disc onto the region bounded by the Cardioid domain. Using a technique introduced by Miller and Mocanu, we investigate several Briot-Bouquet differential subordinations for this function class.

Figures (1)

• Figure :

Theorems & Definitions (15)

• Definition 1.1
• Lemma 1.2
• Lemma 1.3
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 3.1