Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. In this paper, we propose a new fractional identity for multiplicatively differentiable functions; based on this identity, we establish some new fractional multiplicative Bullen-type inequalities for multiplicative differentiable convex functions. Some applications of the obtained results are given.

Multiplicative calculus, also called non-Newtonian calculus, represents an alternative approach to the usual calculus of Newton (1643–1727) and Leibniz (1646–1716). This type of calculus was first introduced by Grossman and Katz and it provides a defined calculation, from the start, for positive real numbers only. In this investigation, we propose to study symmetrical fractional multiplicative inequalities of the Simpson type. For this, we first establish a new fractional identity for multiplicatively differentiable functions. Based on that identity, we derive new Simpson-type inequalities for multiplicatively convex functions via fractional integral operators. We finish the study by providing some applications to analytic inequalities.

In this article, the fractional–space stochastic (2+1)-dimensional breaking soliton equation (SFSBSE) is taken into account in the sense of M-Truncated derivative. To get the exact solutions to the SFSBSE, we use the modified F-expansion method. There are several varieties of obtained exact solutions, including trigonometric and hyperbolic functions. The attained solutions of the SFSBSE established in this paper extend a number of previously attained results. Moreover, in order to clarify the influence of multiplicative noise and M-Truncated derivative on the behavior and symmetry of the solutions for the SFSBSE, we employ Matlab to plot three-dimensional and two-dimensional diagrams of the exact fractional–stochastic solutions achieved here. In general, a noise term that destroy the symmetry of the solutions increases the solution’s stability.

In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By the virtue of the Faber polynomial expansions, the estimation of $n-th$$(n\ge 3)$ Taylor–Maclaurin coefficients $\left|a_n\right|$ is obtained. Furthermore, the bounds value of the first two coefficients of such functions is established.

The notion of a fuzzy subset is used to introduce certain subclasses of analytic functions. Mainly, this article presents several inclusion results and integral preserving properties. Also, certain applications of the analytic functions in terms of fuzzy structure will be discussed.

In this paper, we derive a number of interesting results concerning subordination and superordination relations for certain analytic functions associated with an extension of the Mittag–Leffler function.