Certain classes of multivalent functions defined with higher-order derivatives

Abstract

In this paper we derive some properties of multivalent functions belonging to the classes Rp,q(α) , Bp,q(α) , and Mp,q(α). The results obtained generalize the related works of some authors, and many other new results are obtained.

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

“…More particularly, we write down here that those complex valued functions, which have the forms given as in (1) and also play a big part in the theory of complex functions, are also referred by certain different names in the published literature, which are called as the multivalently (or, r-valently) analytic functions with positive coefficients when all coefficients (w 1+r ) are positive real numbers and the multivalently (or, r-valently) analytic functions with negative coefficients when all coefficients (w 1+r ) are negative real numbers. For all types of these complex functions called as just above, some of the earlier investiga-tions, given in [1,3,5,10,11,13,20,21,25,27], may be presented to the attention of the concerned researchers as certain examples.…”

“…Recently, as it has been informed in details by the investigations in [1,2,4,7,10,11,12,17,18,20,21,22,27,28,29], various types of the multivalently analytic functions with complex coefficients defined by making use of certain elementary operators may also play a significant roles for both operator theory and analytic function theory. Consequently, for those and our scopes, there is in need of two essential reminders in relation with some elementary operators which were earlier constituted by using the multivalently analytic functions being of the forms as in (1). See, especially, the results in the papers presented by [2,11,15].…”

“…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.…”

“…If we want to exemplify here, it is enough to take the value of the parameter v as v := 0. Since Υ (v) (z) is the vth derivative of any multivalently analytic function Υ(z) specified by (1), for all admissible values of those parameters specified as in (4) and also for any multivalently analytic functions with complex coefficients like the form in (1), only one of those special relationships:…”

“…where z ∈ U and the related function Υ := Υ(z) is constituted by the form like (1). By take cognizance of vth derivative operator in (3) for the result in ( 16) and, thereby, after some extensive complex calculations, in a similar ways which were earlier considered in [16], and also under the conditions presented by (15), for the proof of the related theorem, let us then consider the following function Ω(z) given by the implicit form in:…”