Studies on a new K-symbol analytic functions generated by a modified K-symbol Riemann-Liouville fractional calculus.

Analytic functions are very helpful in many mathematical and scientific uses, such as complex integration, potential theory, and fluid dynamics, due to their geometric features. Particularly conformal mappings are widely used in physics and engineering because they make it possible to convert complex physical issues into simpler ones with simpler answers. We investigate a novel family of analytic functions in the open unit disk using the K-symbol fractional differential operator type Riemann-Liouville fractional calculus of a complex variable. For the analysis and solution of differential equations containing many fractional orders, it offers a potent mathematical framework. There are ongoing determinations to strengthen the mathematical underpinnings of K-symbol fractional calculus theory and investigate its applications in various fields.•Normalization is presented for the K-symbol fractional differential operator. Geometric properties are offered of the proposed K-symbol fractional differential operator, such as the starlikeness property and hence univalency in the open unit disk.•The formula of the Alexander integral involving the proposed operator is suggested and studied its geometric properties such as convexity.•Examples are illustrated to fit our pure result. Here, the technique is based on the concepts of geometric function theory in the open unit disk, such as the subordination and Jack lemma.