The study of operators plays an important role in Geometric Function Theory. Many differential
and integral operators can be written in terms of the convolution of certain analytic functions. It
is observed that this formalism makes further mathematical exploration easier, and also improves
the understanding of the geometric and symmetric properties of such operators. The Borel distribution series has applications in geometric function theory, particularly in the
study of univalent functions, coefficient problems, and growth and distortion theorems. The Borel
distribution series can be used to analyze the behavior of these functions, particularly in determining
the distribution of their coefficients.It helps in understanding the probabilistic distribution of these
coefficients, providing insights into their expected magnitude and variance.The Borel distribution
series is used in modeling random holomorphic functions, which are of interest in various areas of
mathematical physics and probability theory. This approach provides insights into the typical behavior of such functions, which can then be applied to specific problems in geometric function theory.
In some cases, the Borel distribution series is applied to study the analytic continuation of functions.
This involves understanding how functions defined by a power series can be extended beyond their
radius of convergence, a topic closely related to the geometric properties of these functions. we obtain geometric properties like Coefficient inequalities, closure theorem, extreme
points and radius of starlikeness and covexity of functions belonging to the subclass.