Development of the Generalized C Operator: Characteristics, Inequalities and Examples

Multi-dimensional integral operators play a crucial role in analysis by enabling the comprehensive evaluation and characterization of complex functions and systems in multiple dimensions. In this setting, the C operator has recently emerged with the distinct features of being nonlinear with an original integral ratio form and having non-linear properties, tractable series expansions and partial derivatives, manageable lower and upper bounds (including Cauchy-Schwarz-type and Lipschitz-type inequalities), convex characteristics, and closed-form expressions for a plethora of functions. In particular, it is connected with a wide panel of well-known integral operators, including the Laplace transform, exponential integral operator, logarithimic integral operator, etc. This article introduces a new multi-dimensional nonlinear operator that expands the capabilities of the C operator. The main novelties in its construction are: (i) a shape parameter that modulates or vanishes the ratio term of the former C operator; and (ii) a multi-dimensional multiplicative power function, also depending on a shape parameter. Due to these novelties, the generalized C operator can be viewed as a weighted version of the C operator. We mainly investigate its notable properties, such as various scale properties, tractable series expansions and partial derivatives, sharp lower and upper bounds, which include ordering comparisons with the original C operator, and convex characteristics. The generalized C operator applied to a few specific functions is provided; some of these represent novel integral results from the literature. Overall, our theoretical findings emphasize the benefits and potential of this new multi-dimensional nonlinear integral operator.