Bi-Univalent Function Classes Defined by Using an Einstein Function and a New Generalised Operator

Abstract

Let A be the class of all analytic and univalent functions f (z) = z+Σ∞k=2 akzk in the open unit disc D = {z:|z|<1 }. S then represents the classes of every function in A that is univalent in D. For every f ∈ S, there is an inverse f−1. A function f ∈ A in D is categorised as bi-univalent if f and its inverse g = f−1 are both univalent. Motivated by the generalised operator, subordination principle, and the first Einstein function, we present a new family of bi-univalent analytic functions on the open unit disc of the complex plane. The functions contained in the subclasses are used to account for the initial coefficient estimate of |a2|. In this study, we derive the results for the covering theorem, distortion theorem, rotation theorem, growth theorem, and the convexity radius for functions of the class Ns,m,kλ,α (Σ, E) of bi-univalent functions related to an Einstein function and a generalised differential operator Ds,m,kλ,α f (z). We use the elementary transformations that preserve the class Ns,m,kλ,α (Σ, E) in order to attain the intended results. The required properties