We introduce and investigate q-analogue of a new subclass of Salagean-type harmonic univalent functions defined by subordination. We first obtained a coefficient characterization of these functions. We give necessary and sufficient convolution conditions, distortion bounds, compactness and extreme points for this subclass of harmonic univalent functions with negative coefficients.

In this paper, we scrutinize some fundamental features of a subclass of harmonic functions defined by a new operator. Like coefficient inequalities, convex combinations, distortion bounds.

We preface and examine classes of (p, q)-convex harmonic locally univalent functions associated with subordination. We acquired a coefficient characterization of (p, q)-convex harmonic univalent functions. We give necessary and sufficient convolution terms for the functions we will introduce.

The study of operators plays an essential role in Mathematics, especially in Geometric Function Theory in Complex Analysis and its related fields. Many derivative and integral operators can be written in terms of convolution of certain analytic functions. The class of analytic functions, which has an essential place in the theory of geometric functions, has been studied by many researchers before. This topic still maintains its popularity today. In this paper, we investigate certain subclasses of analytic functions defined by generalized differential operators involving binomial series. Also, we obtain coefficient estimates involving of the nonhomogeneous Cauchy-Euler differential equation of order .

Harmonic functions are a classic title in the class of geometric functions. Many researchers have studied these function classes from past to present, and since it has a wide range of applications, it is still a popular class. In this study, we will examine harmonic univalent functions, a subclass of harmonic functions. In this study, a subclass of harmonic univalent functions will be examined. Let 𝐻 denote the class of continuous complex-valued harmonic functions which are harmonic in the open unit disk 𝑈 = {𝑧 𝜖 ℂ ∶ |𝑧| < 1} and let 𝐴 be the subclass of 𝐻 consisting of functions which are analytic in 𝑈. A function harmonic in 𝑈 may be written as 𝑓 = ℎ + 𝑔, where ℎ and 𝑔 are analytic in 𝑈. We call ℎ the analytic part and 𝑔 co-analytic part of 𝑓. A necessary and sufficient condition for 𝑓 to be locally univalent and sense-preserving in 𝑈 is that |ℎ′(𝑧)| > |𝑔′(𝑧)| (see [3]). Throughout this paper, we will use introductory notations and delineations of the (p, q)-calculus. The aim of the present paper is to find connections between (p,q)-starlike harmonic univalent functions involving (p,q)-Poisson distribution series.