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.
Highlights
We begin by introducing the important classes of functions considered in this article
A common problem in complex analysis is to determine the range of a function p ∈ H[a, n] from a differential inclusion or containment relation involving several of the derivatives of p
There are many papers of this type that deal with special differential inclusions implying an inclusion for the image of the function p
Summary
We begin by introducing the important classes of functions considered in this article. A common problem in complex analysis is to determine the range of a function p ∈ H[a, n] from a differential inclusion or containment relation involving several of the derivatives of p. There are many papers of this type that deal with special differential inclusions implying an inclusion for the image of the function p. There are many papers that deal with special differential containments and corresponding containments for the image of the function p of the form. We have a differential containment ⇒ function containment Both sets of papers have resulted in many applications in complex analysis. An open question to consider is to combine the two concepts in (1) and (2) and determine conditions on D, Ω and ∆ so that the mixed problem of differential inclusions implies a function containment of the form. It is our intention to do the same with (3)
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