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The main purpose of this work is to study the third Hankel determinant for classes of Bernoulli lemniscate-related functions by introducing new subclasses of star-like functions represented by S L λ * and R L λ . In many geometric and physical applications of complex analysis, estimating sharp bounds for problems involving the coefficients of univalent functions is very important because these coefficients describe the fundamental properties of conformal maps. In the present study, we defined sharp bounds for function-coefficient problems belonging to the family of S L λ * and R L λ . Most of the computed bounds are sharp. This study will encourage further research on the sharp bounds of analytical functions related to new image domains. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]