Fractional Chebyshev-Type Inequalities for Synchronous Functions via Unified Fractional Remainder Operator

In this paper, we establish new fractional Chebyshev-type inequalities for synchronous functions by means of a unified fractional remainder operator. Motivated by the classical Chebyshev integral inequality and its extensions, we investigate how such inequalities can be formulated within a generalized fractional integral framework. To this end, we introduce a normalized form of the unified fractional remainder operator and define an associated fractional Chebyshev functional. Using the nonnegativity and normalization properties of the corresponding fractional kernel, we derive a double-integral representation of this functional. This representation provides the key tool for establishing a fractional analogue of the classical two-function Chebyshev integral inequality for synchronous functions. Furthermore, we extend the approach to obtain a multi-function Chebyshev-type inequality within the same fractional framework. The assumptions ensuring the integrability of the functions and their products are taken into account in establishing these results. Finally, we demonstrate that the classical Chebyshev integral inequality is recovered as a particular case through an appropriate choice of the parameters of the unified operator. These findings extend the applicability of Chebyshev-type inequalities to the proposed fractional setting and provide a framework for deriving related inequalities. Mathematics Subject Classification: 26A33; 26D07; 26D10; 33B15.