Methodology

The MINIMOD project is based on the application and development of modern mathematical methods through three thematically linked work packages. The innovativeness of the research is reflected in the combination of interpolation approaches, functional analysis, geometric constructions and the theory of modular forms.

RP1 – Interpolation methods and generalizations of classical inequalities:
Hermite and Lidstone interpolation polynomials will be applied to generate generalized expressions of Hardy's and Jensen's inequalities. Green's functions will be used to analyze the limits and behavior of functionals over different classes of functions. By combining the Jensen and Lah-Ribaric inequalities, chained integral inequalities over sets from the measure space will be constructed. The conditions under which interpolated expressions maintain convexity and monotonicity will also be investigated.

RP2 – Inequalities and special sequences:
In the second work package, the methodology includes the application of functional analysis of isotonic linear functionals, as well as the analysis of convex functions defined on multidimensional sets. The Jensen–McShane inequality and derived variants of the Diaz–Metcalf and Petrović types will be generalized. Combined inequalities for functions of two variables will also be developed using new types of functionals. An innovative contribution includes their application to the Fibonacci sequence to investigate its structure through the prism of Cauchy and Chebyshev-type inequalities.

RP3 – Geometric structures and modular forms:
The methodology is based on a combination of synthetic and analytical methods. Trilinear coordinates will be used to study families of triangles and develop higher-order curves. Methods for the construction and classification of curves in Cayley–Klein projective planes will be developed. An innovative contribution involves linking geometric structures with modular forms and number theory. Connections will be established between symmetries in geometric configurations and discrete transformations that are relevant to applications in combinatorics and cybersecurity.