Partial sums of analytic functions defined by binomialdistribution and negative binomial distribution

The study of statistical distributions in a complex variable is one of
the most vibrant areas of research. The complex analogue of many
distributions has been studied. This article introduces the complex
analogue of Binomial distribution and Negative Binomial distribution and studies certain geometrical properties of these analogues.
It comprises the study of analytic functions defined by using the
complex versions of Binomial distribution and Negative Binomial distribution functions. It includes the problems of finding the lower
bounds of real parts of certain ratios of partial sums to the infinite
series sums for these analytic functions. Such lower bounds are determined for the said analytic functions, for their first derivatives and for
the Alexander transformation of these analytic functions.