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In this paper, by using Poisson distribution probability, some characteristics of analytic multivalent q-symmetric starlike and q-symmetric convex functions of order η are examined. Then, by utilizing the Poisson distribution and the concept of the q-analogue Salagean integral operator, the p-valent convergence polynomial was introduced. Furthermore, a number of subclasses of analytic symmetric p-valent functions linked to novel polynomials are also deduced. After that, specific coefficient constraints are determined and symmetric δ,q-neighborhoods for p-valent functions are defined. In relation to symmetric δ,q-neighborhoods of q-symmetric p-valent functions formed by Poisson distributions, this paper presents new inclusion results. In addition, a detailed discussion of certain q-symmetric inequalities of analytic functions with negative coefficients is also provided.

In this paper, we employ a q-Noor integral operator to perform a q-analogue of certain fractional integral operator defined on an open unit disc. Then, we make use of the Hadamard convolution product to discuss several related results. Also, we derive a class of convex functions by utilizing the q-fractional integral operator and apply the inspired presented theory of the differential subordination, to geometrically explore the most popular differential subordination properties of the aforementioned operator. In addition, we discuss an exciting inclusion for the given convex class of functions. Over and above, we investigate the q-fractional integral operator and obtain some applications for the differential subordination.

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 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.

Abstract. In this paper, we define and investigate a new class of spirallike harmonic functions defined by a Salagean differential operator and we obtain a coefficient inequality for the functions in this class. Following, we investigated convolution and obtain the order of convolution consistence for certain spirallike harmonic univalent functions with negative coefficients.

In this paper, we define a differential operator on an open unit disk Δ by using the novel Borel distribution (BD) operator and means of convolution. This operator is adopted to introduce new subclasses of p-valent functions through the principle of differential subordination, and we focus on some interesting inclusion relations of these classes. Additionally, some inclusion relations are derived by using the Bernardi integral operator. Moreover, relevant convolution results are established for a class of analytic functions on Δ, and other results of analytic univalent functions are derived in detail. This study provides a new perspective for developing p-univalent functions with BD and offers valuable understanding for further research in complex analysis.

In this paper, we discuss stochastic comparisons of active redundancy at component level versus system level. We also consider series systems in order to compare their lifetimes using the usual stochastic, the hazard rate and the reversed hazard rate orders, for two cases: (i) the spare and parent components are independent and have the same distribution, and then (ii) the spare and parent components are dependent and have not the same distribution. ./files/site1/files/72/6Abstract.pdf

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.