In this paper we obtain some applications of first order differential subordination and superordination result involving an integral operator for certain normalized analytic function.

<abstract><p>Using the (p, q)-derivative operator we introduce new subclasses of analytic and bi-univalent functions, we obtain estimates on coefficients and the Fekete-Szegö functional.</p></abstract>

The results presented in this paper deal with the classical but still prevalent problem of introducing new classes of m-fold symmetric bi-univalent functions and studying properties related to coefficient estimates. Quantum calculus aspects are also considered in this study in order to enhance its novelty and to obtain more interesting results. We present three new classes of bi-univalent functions, generalizing certain previously studied classes. The relation between the known results and the new ones presented here is highlighted. Estimates on the Taylor–Maclaurin coefficients |am+1| and |a2m+1| are obtained and, furthermore, the much investigated aspect of Fekete–Szegő functional is also considered for each of the new classes.

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 AE(λ,σ,δ,s,t,p,q;ϑ) and SE(μ,γ,σ,δ,s,t,p,q;ϑ) 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 we introduce a new class of analytic functions f in the open unit disk normalized by f(0) = f′ (0)−1 = 0, associated with exponential functions. The aim of the present paper is to investigate the third-order Hankel determinant H 3(1) for this function class and obtain the upper bound of the determinant H 3(1).

In this paper, we introduce three new subclasses of m-fold symmetric holomorphic functions in the open unit disk U, where the functions f and f−1 are m-fold symmetric holomorphic functions in the open unit disk. We denote these classes of functions by FSΣ,mp,q,s(d), FSΣ,mp,q,s(e) and FSΣ,mp,q,s,h,r. As the Fekete-Szegö problem for different classes of functions is a topic of great interest, we study the Fekete-Szegö functional and we obtain estimates on coefficients for the new function classes.

In the current article, making use of certain operator, we initiate and explore a certain family WΣ(λ,γ,σ,δ,α,β,p,q;h) of holomorphic and bi-univalent functions in the open unit disk D. We establish upper bounds for the initial Taylor–Maclaurin coefficients and the Fekete–Szegö type inequality for functions in this family.

In this paper, we define certain families SE*(ϑ) and CE(ϑ) of holomorphic and bi-univalent functions which are defined in the open unit disk U. We establish upper bounds for the initial Taylor–Maclaurin coefficients and Fekete–Szegö type inequalities for functions in these families.