A note on some relations between certain inequalities and normalized analytic functions

Abstract

In this note, an extensive result consisting of several relations between certain inequalities and normalized analytic functions is first stated and some consequences of the result together with some examples are next presented. For the proof of the presented result, some of the assertions indicated in [5], [8] and [11] along with the results in [3] and [4] are also considered.

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

“…The well-known assertion (Lemma 2.1, below), proven by Nunokawa [19], will be needed to state and also prove main results of this investigation. For both its details and some implications of the earlier results demonstrated by the help of Lemma 2.1, one may refer to [17] and [18], and, as example, also [9,10,21].…”

Some results concerning the Tremblay operator and some of its applications to certain analytic functions

“…As example and special consequences of them, we want to emphasize only two appertaining to the study of the relations between the analytic properties of a function f(z) in the class A(n) and the geometric properties of the image domain f U . One may see their details in [4], [6] and [8]- [12]. The others possible consequences, which are well-known in the literature and are also omitted here, are presented to reveal to the attention of the researchers who are interested in the topics of this work.…”

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

“…which was defined in the domain of the complex plane and whose properties in several spaces were discussed systematically. For both the details of FC and, as example, some novel results proven by certain different methods, one may refer to the scientific works given by [1]- [3], [5], [6]- [13] and see also [14]- [18].…”