Search

In this paper, we present a new class of linear fractional differential operators that are based on classical Gaussian hypergeometric functions. Then, we utilize the new operators and the concept of differential subordination to construct a convex set of analytic functions. Moreover, through an examination of a certain operator, we establish several notable results related to differential subordination. In addition, we derive inclusion relation results by employing Briot–Bouquet differential subordinations. We also introduce a perspective study for developing subordination results using Gaussian hypergeometric functions and provide certain properties for further research in complex dynamical systems.

In this paper, we make use of the Riemann–Liouville fractional q-integral operator to discuss the class Sq,δ*(α) of univalent functions for δ>0,α∈C−{0}, and 0<|q|<1. Then, we develop convolution results for the given class of univalent functions by utilizing a concept of the fractional q-difference operator. Moreover, we derive the normalized classes Pδ,qζ(β,γ) and Pδ,q(β) (0<|q|<1, δ≥0,0≤β≤1,ζ>0) of analytic functions on a unit disc and provide conditions for the parameters q,δ,ζ,β, and γ so that Pδ,qζ(β,γ)⊂Sq,δ*(α) and Pδ,q(β)⊂Sq,δ*(α) for α∈C−{0}. Finally, we also propose an application to symmetric q-analogues and Ruscheweh’s duality theory.

In this paper, we define and discuss properties of various classes of analytic univalent functions by using modified q-Sigmoid functions. We make use of an idea of Salagean to introduce the q-analogue of the Salagean differential operator. In addition, we derive families of analytic univalent functions associated with new q-Salagean and q-Ruscheweh differential operators. In addition, we obtain coefficient bounds for the functions in such new subclasses of analytic functions and establish certain growth and distortion theorems. By using the concept of the (q, δ)-neighbourhood, we provide several inclusion symmetric relations for certain (q, δ)-neighbourhoods of analytic univalent functions of negative coefficients. Various q-inequalities are also discussed in more details.

In the present paper, we discuss a class of bi-univalent analytic functions by applying a principle of differential subordinations and convolutions. We also formulate a class of bi-univalent functions influenced by a definition of a fractional q-derivative operator in an open symmetric unit disc. Further, we provide an estimate for the function coefficients |a2| and |a3| of the new classes. Over and above, we study an interesting Fekete–Szego inequality for each function in the newly defined classes.