A new insight into meromorphic functions: An exploration of negative r order of two convex and starlike dynamics

Abstract

Analytical and univalent functions play a critical role in the study of complex analysis. Specifically, functions that fall under the Sand ∑classes exhibit unique characteristics, making them important subjects of study. In this research, we focus on these specific classes and utilize certain inherent properties associated with them. A fundamental area of interest is the open D = {z : 0 < |z| < 1,z ∈ C}unit disk in the complex plane. In this domain, our primary focus is on understanding the behavior of these functions under specific conditions, namely, logarithmic derivative conditions. Logarithmic derivatives are an essential tool in determining the nature and behavior of a function within its domain. In this context, we have been successful in deriving 2−r ( r=1,2,3,..) order special starlike and convex functions. Starlike functions are a subset of univalent functions that exhibit a specific shape-preserving property, while convex functions are those for which the line segment between any two points on the graph of the function lies above or on the graph itself. By leveraging certain properties of the Sand ∑classes and applying the logarithmic derivative conditions within the open D unit disk, this research provides new insights and results into the study of 2−r -order special starlike and convex functions.