Analysis of a Normalized Structure of a Complex Fractal–Fractional Integral Transform Using Special Functions

• Published in: Axioms Vol. 13; no. 8; p. 522
• Main Authors: Ibrahim, Rabha W., Salahshour, Soheil, Páll-Szabó, Ágnes Orsolya
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-08-2024
• Subjects: Analytic functions; complex transform; Complex variables; Fractal analysis; fractal calculus; Fractal transforms; Fractals; fractal–fractional calculus; fractal–fractional differential operator; Fractional calculus; fractional difference operator; Functional equations; Functions; Gamma function; Integral transforms; Integrals; Laplace transforms; Mathematical analysis; Operators (mathematics); Power series; Turkey
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms13080522

By using the most generalized gamma function (parametric gamma function, or p-gamma function), we present the most generalized Rabotnov function, called the p-Rabotnov function. Consequently, new fractal–fractional differential and integral operators of a complex variable in an open unit disk are defined and investigated analytically and geometrically. We address some inequalities involving the generalized fractal–fractional integral operator in some spaces of analytic functions. A novel complex fractal–fractional integral transform (CFFIT) is presented. A normalization of the proposed CFFIT is observed in the open unit disk. Examples are illustrated for power series of analytic functions.