Majorization properties for analytic functions with positive real part

The majorization feature is important when investigating analytical functions’ geometric properties. The present work focuses on the majorization property for a general Ma-Minda type function class S*φ(γ, λ) of starlike functions of complex order, which is described with the Ruscheweyh differential operator. In the context of the main theorem, consequences are presented for various special functions φ having a positive real part. Furthermore, the Ruscheweyh differential operator extends a representation formula and investigates an upper-bound formula. Examples of upper bounds for analytic functions are also presented.