Applications of Certain Conic Domains to a Subclass of q-Starlike Functions Associated with the Janowski Functions

Abstract

In our present investigation, with the help of the basic (or q-) calculus, we first define a new domain which involves the Janowski function. We also define a new subclass of the class of q-starlike functions, which maps the open unit disk U, given by U= z:z∈C and z <1, onto this generalized conic type domain. We study here some such potentially useful results as, for example, the sufficient conditions, closure results, the Fekete-Szegö type inequalities and distortion theorems. We also obtain the lower bounds for the ratio of some functions which belong to this newly-defined function class and for the sequences of the partial sums. Our results are shown to be connected with several earlier works related to the field of our present investigation. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward (p,q)-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter p is obviously redundant.

A Subclass of Multivalent Janowski Type q-Starlike Functions and Its Consequences

“…Moreover, the subclasses of q-starlike functions associated with the Janwoski or some other functions have been studied by the many authors (see, for example, [13][14][15][16][17][18][19][20]). For some more recent investigations based upon the q-calculus, we may refer the interested reader to the works in [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. Our present research is a continuation of some of these earlier developments.…”

“…In this article, we are essentially motivated by the recently published paper of Khan et al in Symmetry (see [27]) and some other related works on this subject, which we have mentioned above. We first introduce a new subclass of analytic and multivalent (or p-valent) functions by using the concept of the q-calculus in association with the Janowski functions.…”

A certain q-Ruscheweyh type derivative operator and its applications involving multivalent functions

“…For details, we refer the reader to the earlier works [1,7,18,21,23]. Moreover, several authors (see, for example, [13,19,42,47,48]) have concentrated upon the classes of q-starlike functions related with the Janowski and other functions from several different viewpoints and the references cited therein. For some more recent investigations involving q-calculus, we may refer the interested reader to [4, 5, 12, 14-17, 20, 22, 26-28, 31, 34-36, 38, 43, 46].…”

A Class of k-Symmetric Harmonic Functions Involving a Certain q-Derivative Operator

“…Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials are applicable particularly in several diverse areas of mathematical and physical sciences (see, for example, [41] (pp. 350-351); see also [42][43][44][45][46][47][48]). Moreover, as we remarked above and in the introductory Section 1, in Srivastava's recently-published survey-cum-expository review article [12], the triviality of the so-called (p, q)-calculus was exposed and it also mentioned about the trivial and inconsequential variation of the classical q-calculus to the so-called (p, q)calculus, the additional parameter p being redundant or superfluous see, for details, [12] (p. 340) .…”