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:
$$\mathcal{S}_{\rm starlike}^{\alpha,\lambda,q}(b,A,B)
\qquad \text{and} \qquad
\mathcal{S}_{\rm convex}^{\alpha,\lambda,q}(b,A,B)$$
of starlike and convex functions in the open unit disk $\Delta$
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
$\;\mathcal{S}_{\rm starlike}^{\alpha,\lambda,q}(b,A,B)\;$
and $\;\mathcal{S}_{\rm convex}^{\alpha,\lambda,q}(b,A,B)\;$
with the function classes which were investigated in several
earlier works.