<![CDATA[This study aims to explore the geometric properties of the generalized probability distribution associated with the generalized Koebe function. The inequality coefficients in the convoluted power series are evaluated using Chebyshev polynomials, with a focus on determining bounds on the initial coefficients. This research employs a mathematical-theoretic approach utilizing algebraic manipulations, theorem proving, and analytical techniques such as subordination and Hadamard products. Few conditions were set on some variables which yields certain corollaries. The results suggest that the analyzed functions have properties that are applicable in modeling complex data, allowing the description of non-normal, asymmetric, or heavy-tailed probability distributions. These findings corroborate the relevance of generalized distributions in dealing with uncertainty and variability in statistical data. Based on the existing works, the results obtained in this work are just derived. In conclusion, this study contributes to understanding the geometric relationships of generalized probability distributions. The practical implications of these findings include applications in the analysis of complex statistical data and modeling of real-world phenomena that require non-conventional distributions.]]>
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