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In this paper, we employ the concept of the $ q $-derivative to derive certain differential and integral operators, $ D_{q, \lambda}^{n} $ and $ I_{q, \lambda}^{n} $, resp., to generalize the class of Salagean operators over the set of univalent functions. By means of the new operators, we establish the subclasses $ M^n_{q, \lambda} $ and $ D^n_{q, \lambda} $ of analytic functions on an open unit disc. Further, we study coefficient inequalities for each function in the given classes. Over and above, we derive some properties and characteristics of the set of differential subordinations by following specific techniques. In addition, we study the general properties of $ D_{q, \lambda}^{n} $ and $ I_{q, \lambda}^{n} $ and obtain some interesting differential subordination results. Several results are also derived in some details.

In this article we continue our investigation of the Paatero space. We prove that the intersection of every Paatero class V(k) with every Hardy space Hp is closed in that Hp and associate singular continuous measures to elements of V(k). There have been no examples in the literature of functions in V(k) with zeros in the unit disk other than the one at the origin. We close this gap in the literature. We derive a representation of the measure associated to a function in V(k) for functions whose derivatives are rational, or algebraic, or transcendental functions in the unit disk.Finally, we consider the notion of regulated domains, introduced by Pommerenke and show that there are regulated domains whose boundary is not of bounded boundary rotation.

The Gould–Hopper–Laguerre–Sheffer matrix polynomials were initially studied using operational methods, but in this paper, we investigate them using matrix techniques. By leveraging properties of Pascal functionals and Wronskian matrices, we derive several identities for these polynomials, including recurrence relations. It is highlighted that these identities, acquired via matrix techniques, are distinct from the ones obtained when using operational methods.

This article presents certain families of analytic functions regarding q-starlikeness and q-convexity of complex order γ ( γ ∈ C \ 0 ) . This introduced a q-integral operator and certain subclasses of the newly introduced classes are defined by using this q-integral operator. Coefficient bounds for these subclasses are obtained. Furthermore, the ( δ , q )-neighborhood of analytic functions are introduced and the inclusion relations between the ( δ , q )-neighborhood and these subclasses of analytic functions are established. Moreover, the generalized hyper-Bessel function is defined, and application of main results are discussed.