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

Abstract

The aim of this scientific note is first to present certain information associated with the Tremblay operator in the complex plane and then to determine several results constituted by the related operator for certain analytic functions and also to point some implications of them out. 2010 Mathematics Subject Classification: 26A33, 34A08, 35G10, 35F05, 30C55, 30C45, 30A10 Key words and phrases: complex plane, unit open disk, normalized analytic function, fractional calculus, tremblay operator, equations and inequalities in the complex plane

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.…”

“…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]. Through the instrumentality of the series expansions given in (1), (5) and 6, under the conditions of the admissible values of the parameters indicated in (10) [196]…”

“…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.…”

“…Through the instrument of the definition in (3), one of the operators is also the Tremblay operator, which was defined in the domain of the complex plane and whose properties in several spaces were discussed systematically by several researchers for their researches. For the details one may refer to [1,2,3,7,8,14,15,16,18,19,20,22].…”