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

Bilal Khan,Muhammad Tahir,Qazi Zahoor Ahmad,Nazar Khan,Hari Mohan Srivastava,Maslina Darus

doi:10.3390/sym13040574

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.

Highlights

In our application based upon the above definition, we introduce and study the corresponding q-extension of the function class k-S ∗ [ A, B] as follows
We aim to investigate the Fekete-Szegö functional a3 − μa22 for the class S ∗ (q, k, A, B) of Janowski type q-starlike functions which is associated with a certain conic domain
Let T be a subset of the normalized analytic function class A consisting of functions with negative TaylorMaclaurin coefficients, that is

Summary

Sufficient Conditions

This section is devoted to the study of sufficient conditions for a function f to be in the class S ∗ (q, k, A, B). A normalized analytic function f having the series expansion given in (1) is placed in the class S ∗ (q, k, A, B) if the following condition holds true:. (see [4]) A normalized analytic function f having series expansion given in (1) is in the class k-ST if the following condition holds true:. (see [9]) A normalized analytic function f having series expansion given in (1) is in the class SD(k, α) if it satisfies the following condition:. ∑ {n(k + 1) − (k + α)}|an | < (1 − α)

Closure Theorems

The Fekete-Szegö Functional

Analytic Functions with Negative Coefficients

Concluding Remarks and Observations