Exploring n-Trigonometrically Convexity and Hermite-Hadamard Type Inequalities
DOI:
https://doi.org/10.62810/jnsr.v4i3.474Keywords:
Convex function, Hermite-Hadamard inequality, n-Trigonometrically convex function, Special meansAbstract
In this paper, we introduce and systematically investigate n-trigonometrically convex functions, providing a rigorous definition and discussing their fundamental properties. We examine several algebraic and analytical properties of this newly defined class of functions in detail, revealing their structural behavior and relationships with existing convexity notions. To analyze this class, we use a methodological combination of trigonometric coefficients and polynomial structures. As a central contribution, we establish Hermite–Hadamard-type inequalities for n-trigonometrically convex functions, extending the well-known classical results for standard convex functions. Furthermore, we derive a variety of new integral inequalities for functions whose first derivative, in absolute value and raised to a power greater than one or at least one, is n-trigonometrically convex. To establish these new analytical bounds, we use a rigorous methodology based on novel integral identities, which we evaluate using the classical Hölder and power-mean integral inequalities. These inequalities provide refined bounds and generalizations that enrich the current theory of convex analysis. The obtained results not only broaden the scope of classical convexity concepts but also introduce new analytical tools that may be applied in different areas of mathematical analysis and inequality theory. In the final part of the paper, we demonstrate the applicability of the established results by applying them to several special means of real numbers, including well-known means such as the arithmetic and geometric means. These applications illustrate the practical relevance and effectiveness of the proposed theory and highlight its potential for further developments in both theoretical and applied mathematics.
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Copyright (c) 2026 Mohammad Mubarak, Abdul Wakil Baidar, Noorullah Noori
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