In this article, by making use of the q-Srivastava-Attiya operator, we introduce and investigate a new family ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$ of normalized holomorphic and bi-univalent functions in the open unit disk $\mathbb{U}$, which are associated with the Bazilevič functions and the $\lambda$-pseudo-starlike functions as well as the Horadam polynomials. We estimate the second and the third coefficients in the Taylor-Maclaurin expansions of functions belonging to the holomorphic and bi-univalent function class, which we introduce here. Furthermore, we establish the Fekete-Szegö inequality for functions in the family ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$. Relevant connections of some of the special cases of the main results with those in several earlier works are also pointed out. Our usage here of the basic or quantum (or q-) extension of the familiar Hurwitz-Lerch zeta function $\Phi (z,s,a)$ is justified by the fact that several members of this family of zeta functions possess properties with local or non-local symmetries. Our study of the applications of such quantum (or q-) extensions in this paper is also motivated by the symmetric nature of quantum calculus itself.

The purpose of the present paper is to determine the necessary and sufficient conditions for the power series B_{\mu} whose coefficients are probabilities of the Borel distribution to be in the family H(\lambda, \sigma, \delta, \mu) of analytic functions which defined in the open unit disk. We derive a number of important geometric properties, such as, coefficient estimates, integral representation, radii of starlikeness and convexity. Also we discuss the extreme points and neighborhood property for functions belongs to this family.

In the present investigation, we use the Horadam Polynomials to establish upper bounds for the second and third coefficients of functions belongs to a new subclass of analytic and Bazilevic bi-univalent functions defined in the open unit disk ∪. Also, we discuss Fekete-Szegä problem for functions belongs to this subclass.2010 Mathematics Subject Classification: 30C45, 30C50.

The motivation of the present article is to define the $(p-q)$-Wanas operator in geometric function theory by the symmetric nature of quantum calculus. We also initiate and explore certain new families of holormorphic and bi-univalent functions ${\mathcal{A}}_E(\lambda ,\sigma ,\delta ,s,t,p,q;\vartheta )$ and $S_E(\mu ,\gamma ,\sigma ,\delta ,s,t,p,q;\vartheta )$ which are defined in the unit disk U associated with the $(p-q)$-Wanas operator. The upper bounds for the initial Taylor–Maclaurin coefficients and Fekete–Szegö-type inequalities for the functions in these families are obtained. Furthermore, several consequences of our results are pointed out based on the various special choices of the involved parameters.

In the present paper, due to beta negative binomial distribution series and Laguerre polynomials, we investigate a new family ${\mathcal{F}}_{\mathsf{\Sigma }}(\delta ,\eta ,\lambda ,\theta ;h)$ of normalized holomorphic and bi-univalent functions associated with Ozaki close-to-convex functions. We provide estimates on the initial Taylor–Maclaurin coefficients and discuss Fekete–Szegő type inequality for functions in this family.