| Summary: | Let A be the class of all analytic and univalent functions (formula presented) in the open unit disc (formula presented). 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 (formula presented) of bi-univalent functions related to an Einstein function and a generalised differential operator (formula presented)We use the elementary transformations that preserve the class (formula presented) in order to attain the intended results. The required properties are then obtained. © 2023 The Authors.
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