Systems of Simultaneous Differential Inclusions Implying Function Containment

Abstract

An important problem in complex analysis is to determine properties of the image of an analytic function p defined on the unit disc U from an inclusion or containment relation involving several of the derivatives of p. Results dealing with differential inclusions have led to the development of the field of Differential Subordinations, while results dealing with differential containments have led to the development of the field of Differential Superordinations. In this article, the authors consider a mixed problem consisting of special differential inclusions implying a corresponding containment of the form D[p](U)⊂Ω⇒Δ⊂p(U), where Ω and Δ are sets in C, and D is a differential operator such that D[p] is an analytic function defined on U. We carry out this research by considering the more general case involving a system of two simultaneous differential operators in two unknown functions.

Keywords

• differential inclusions
• differential containments
• differential inequalities
• differential subordinations
• univalent functions