The paper introduces a novel definition of tri-topological spaces, extending the classical theory of bi-topological spaces as developed by Kelly. It presents a unified framework linking separation axioms with compactness properties, building on results from Willard and Engelking. Central to this work is the interaction function $\rho :{\tau }_1\times {\tau }_2\times {\tau }_3\to \mathcal{P}(\mathcal{P}(X))$, which encodes complex relationships among three topologies and satisfies five key axioms (TT1–TT5). This enables the modeling of topological phenomena beyond simple unions or products. The paper explores connections between tri-topological spaces and Lindelöf, paracompact, metacompact, and connected spaces. Several new theoretical results are presented with complete proofs, and practical relevance is demonstrated in three areas: digital topology, data analysis, and quantum gravity. Overall, the study offers new insights into point-set topology by integrating previously unrelated topological structures.

The research here was motivated by a number of recent studies on Hankel inequalities and sharp bounds. In this article, we define a new subclass of holomorphic convex functions that are related to tangent functions. We then derive geometric properties like the necessary and sufficient conditions, radius of convexity, growth, and distortion estimates for our defined function class. Furthermore, the sharp coefficient bounds, sharp Fekete-Szegö inequality, sharp 2nd order Hankel determinant, and Krushkal inequalities are given. Moreover, we calculate the sharp coefficient bounds, sharp Fekete-Szegö inequality, and sharp second-order Hankel determinant for the functions whose coefficients are logarithmic.

Logarithmic functions are widely used in mathematics and other fields for various applications. As far as we know, no one has used the bounds for the third Hankel determinant using the coefficients of logarithmic functions. This article examined various classes of starlike functions and addressed the problem of the third Hankel determinant concerning logarithmic coefficients for special subclasses related to nephroid functions. Several coefficient estimates were derived, and some of these results were proven to be sharp.

In this paper, we provide the notion of a generalized neutrosophic contraction, which extends the concepts of neutrosophic non-expansive mappings and neutrosophic Banach contractions. Using this approach, we proved a fixed point theorem and demonstrated the existence and uniqueness of a solution in the context of neutrosophic metric space and discussed its importance with some appealing applications such as the satellite web coupling problem. We also investigated a new avenue in fractal function production, where fractal structures are constructed using neutrosophic Hutchinson-Barnsley (NHB) operators. We have provided a variety of very interesting examples to illustrate the efficiency of our work in complicated dynamical systems, fractal geometry, and iterated function systems (IFS). We set the stage for future studies in applied mathematics, stability analysis, and fixed point theory by utilizing neutrosophic contraction principles.