The word “symmetry” is a Greek word that originated from “symmetria”. It means an agreement in dimensions, due proportion, and arrangement; however, in complex analysis, it means objects remaining invariant under some transformation. This idea has now been recently used in geometric function theory to modify the earlier classical q-derivative introduced by Ismail et al. due to its better convergence properties. Consequently, we introduce a new class of analytic functions by using the notion of q-symmetric derivative. The investigation in this paper obtains a number of the latest important results in q-theory, including coefficient inequalities and convolution characterization of q-symmetric starlike functions related to Janowski mappings.

In this paper, we introduce two new subclasses of bi-univalent functions using the q-Hermite polynomials. Furthermore, we establish the bounds of the initial coefficients ${\upsilon }_2$, ${\upsilon }_3$, and ${\upsilon }_4$ of the Taylor–Maclaurin series and that of the Fekete–Szegö functional associated with the new classes, and we give the many consequences of our findings.

Very recently, functions that map the open unit disc U onto a limaçon domain, which is symmetric with respect to the real axis in the right-half plane, were initiated in the literature. The analytic characterization, geometric properties, and Hankel determinants of these families of functions were also demonstrated. In this article, we present a q-analogue of these functions and use it to establish the classes of starlike and convex limaçon functions that are correlated with q-calculus. Furthermore, the coefficient bounds, as well as the third Hankel determinants, for these novel classes are established. Moreover, at some stages, the radius of the inclusion relationship for a particular case of these subclasses with the Janowski families of functions are obtained. It is worth noting that many of our results are sharp.

In the past few years, many scholars gave much attention to the use of q-calculus in geometric functions theory, and they defined new subclasses of analytic and harmonic functions. While using the symmetric q-calculus in geometric function theory, very little work has been published so far. In this research, with the help of fundamental concepts of symmetric q-calculus and the symmetric q-Salagean differential operator for harmonic functions, we define a new class of harmonic functions connected with Janowski functions $\widetilde{{\mathcal{S}}_{\mathcal{H}}^0}\left(m,q,\mathcal{A},\mathcal{B}\right)$. First, we illustrate the necessary and sufficient convolution condition for $\widetilde{{\mathcal{S}}_{\mathcal{H}}^0}\left(m,q,\mathcal{A},\mathcal{B}\right)$ and then prove that this sufficient condition is a sense preserving and univalent, and it is necessary for its subclass $\widetilde{{\mathcal{TS}}_{\mathcal{H}}^0}\left(m,q,\mathcal{A},\mathcal{B}\right)$. Furthermore, by using this necessary and sufficient coefficient condition, we establish some novel results, particularly convexity, compactness, radii of q-starlike and q-convex functions of order $\alpha$, and extreme points for this newly defined class of harmonic functions. Our results are the generalizations of some previous known results.

In this study, we apply a recently developed idea of up and down fuzzy-ordered relations between two fuzzy numbers. Here, we consider fuzzy Riemann–Liouville fractional integrals to establish the Hermite–Hadamard-, Fejér-, and Pachpatte-type inequalities. We estimate fuzzy fractional inequalities for a newly introduced class of ℏ-preinvexity over fuzzy-number valued settings. For the first time, such inequalities involving up and down fuzzy-ordered functions are proven using the fuzzy fractional operator. The stated inequalities are supported by a few numerical examples that will be helpful to validate our main results.

This paper’s main goal is to introduce left and right exponential trigonometric convex interval-valued mappings and to go over some of their important characteristics. Additionally, we demonstrate the Hermite–Hadamard inequality for interval-valued functions by utilizing fractional integrals with exponential kernels. Moreover, we use the idea of left and right exponential trigonometric convex interval-valued mappings to show various findings for midpoint- and Pachpatte-type inequalities. Additionally, we show that the results provided in this paper are expansions of several of the results already demonstrated in prior publications The suggested research generates variants that are applicable for conducting in-depth analyses of fractal theory, optimization, and research challenges in several practical domains, such as computer science, quantum mechanics, and quantum physics.