Fractional Product Inequalities for Special Functions via Generalized Harmonic ψ − s-Convexity and Local Fractional Integrals

This paper develops a unified framework of fractional product inequalities for special functions, extending classical Cauchy-Schwarz type results to nonclassical convexity settings. The approach exploits the structural properties of generalized harmonic Formula: see text-convex functions in combination with local fractional integral operators to derive sharp and explicitly computable bounds. The analysis employs auxiliary identities together with refined Hölder and power-mean techniques adapted to the fractional context. Within this framework, both direct and reverse fractional product inequalities are established, with the deformation mechanisms and coefficient functions characterized in terms of fractional order and convexity parameters. Applications are presented for several important classes of special functions, including Gamma-type, Bessel-type, and hypergeometric-type functions. Moreover, many existing inequalities are recovered as particular or limiting cases, demonstrating the flexibility and generality of the approach. These results provide a broad and practical extension of classical inequality techniques, with potential relevance to fractional calculus, analysis, and related mathematical applications.