Subclasses of analytic functions with respect to symmetric and conjugate points connected with the q-Borel distribution

Abstract

In this article, by making use of a q-analogue of the familiar Borel distribution, we introduce two new subclasses: S?,?,q symmetric(b, A, B) and S?,?,q conjugate(b,A, B) of starlike and convex functions in the open unit disk ? with respect to symmetric and conjugate points. We obtain some properties including the Taylor-Maclaurin coefficient estimates for functions in each of these subclasses and deduce various corollaries and consequences of the main results. We also indicate relevant connections of each of these subclasses S?,?,q symmetric(b,A, B) and S?,?,q conjugate(b,A, B) with the function classes which were investigated in several earlier works. Finally, in the concluding section, we choose to comment on the recent usages, especially in Geometric Function Theory of Complex Analysis, of the basic (or q-) calculus and also of its trivial and inconsequential (p, q)-variation involving an obviously redundant (or superfluous) parameter p.