A note on some elementary properties and applications of certain operators to certain functions analytic in the unit disk

Abstract

This scientific note relates to introducing certain elementary operators defined in the unit disk in the complex plane, then determining various applications (specified by those operators) to certain analytic functions, and also revealing a number of possible implications of them.

A variety of multivalently analytic functions with complex coefficients and some argument properties of their applications

“…The proposition (just below) is an important tool and very useful auxiliary theorem proven by M. Nunokawa (see [24]). For some of the earlier results proven by that assertion (just below), one may refer to the papers given in [14,15,16] as some examples.…”

“…By both considering Lemma 1 and using various extensive combining determined by certain (elementary) operators in (7) (see [15] and also [14,16]), we now can both create and then prove an extensive result consisting of various arguments of certain rational types of the regular functions like (1), which is being the following theorem (below). Theorem 1.…”

“…In the literature, of course, it also is possible to see numerous scientific researches associating with certain special types constituted by some of earlier elementary differential operators (or, their implications / applications) that we have also defined just above. For some of them, there also is a need to call up some comprehensive information (or, reminders) (cf., e.g., [1,2,3,11,13,14,15]), as follows.…”

“…As a second reminder, we also need to start with presenting an extra-elementary operator defined by the help of the definition in [15]. (See also [14] and [16].) For its definition, in the light of the information given by (2)-( 5), for a multivalently analytic function like Υ(z) as in (1), we then establish the following-differential operator given by…”

“…In the present cases, as is well known, these types of the indicated-special relations between the mentioned operators D v φ Υ (z) and Υ (v) (z) also play an important role for (Analytic and) Geometric Function Theory. (See [6,9] for its special topics and see also [6,8,9,14,16,19,23,26], as examples.) If we want to exemplify here, it is enough to take the value of the parameter v as v := 0.…”