Certain basic information related to the Tremblay operator and some applications in connection therewith

Abstract

In this scientific note, an operator, which is the well-known Tremblay operator in the literature, is first introduced and some of its applications to certain analytic complex functions, which are normalized and analytic in the open unit disk, are then determined. In addition, certain special results of the related applications are also emphasized.

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

“…and, under the conditions accentuated in (10), the following-comprehensive relations between the related operators, which were also indicated in [11,12,13,16], can be easily emphasized by the equivalent statements:…”

“…The following-elementary results are only two of several applications of the operator, defined as in (4), which will be needed for proving our main results A note on some properties and applications of certain operators [195] relating to the analytic functions of the form (1). In the same breath, for their details and some of their extensive applications, it can be also checked the papers in [5], and see also the recent works in [11,12,13,16] as certain examples.…”

“…respectively, where z ∈ U. We remark here that, for the contents of the two operators just above, it can be focused on certain earlier results relating to certain analytic functions considered in [11] and [12]. In the same time, certain relations between operators in (11)-(13) and some of their implications can be also compared with some of the results determined by those operators in [13] and [16].…”

“…and is also defined by the definition given in [1,2,3,7,11,12,13,14,16,20]. Let κ(z) be an analytic function in a simply-connected region of the complex plane containing the origin.…”

Geometric properties of some applications of the Tremblay operator

“…For both some of the details of fractional calculus and some applications identified by certain (elementary) operators (in relation with fractional calculus), one may refer to the scientific investigations given by [1], [6] and [10]- [12]. For an analytic function g := g(z) belonging to the class A(n), in general, the Tremblay operator:…”