Both the theory of differential subordination and its dual, the theory of differential superordination, introduced by Professors Miller and Mocanu are based on reinterpreting certain inequalities for real-valued functions for the case of complex-valued functions. Studying subordination and superordination properties using different types of operators is a technique that is still widely used, some studies resulting in sandwich-type theorems as is the case in the present paper. The fractional integral of confluent hypergeometric function is introduced in the paper and certain subordination and superordination results are stated in theorems and corollaries, the study being completed by the statement of a sandwich-type theorem connecting the results obtained by using the two theories.

A non-polynomial spline is a technique that utilizes information from symmetric functions to solve mathematical or physical models numerically. This paper introduces a novel non-polynomial spline construct incorporating a rational function term to develop an efficient numerical scheme for solving time-fractional differential equations. The proposed method is specifically applied to the time-fractional KdV–Burgers (TFKdV) equation. and time-fractional differential equations are crucial in physics as they provide a more accurate description of various complex processes, such as anomalous diffusion and wave propagation, by capturing memory effects and non-local interactions. Using Taylor expansion and truncation error analysis, the convergence order of the numerical scheme is derived. Stability is analyzed through the Fourier stability criterion, confirming its conditional stability. The accuracy and efficiency of the rational non-polynomial spline (RNPS) method are validated by comparing numerical results from a test example with analytical and previous solutions, using norm errors. Results are presented in $2D$ and $3D$ graphical formats, accompanied by tables highlighting performance metrics. Furthermore, the influences of time and the fractional derivative are examined through graphical analysis. Overall, the RNPS method has demonstrated to be a reliable and effective approach for solving time-fractional differential equations.

In this paper, we introduce new subclasses ${\mathfrak{R}}_{\mathsf{\Sigma },b,c}^{\mu ,\alpha }\left(\lambda ,\delta ,\tau ,\mathsf{\Phi }\right)$ and ${\mathfrak{K}}_{\mathsf{\Sigma },b,c}^{\mu ,\alpha }\left(\lambda ,\delta ,\eta ,\mathsf{\Phi }\right)$ of bi-univalent functions in the open unit disk U by using quasi-subordination conditions and determine estimates of the coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for functions of these subclasses. We discuss the improved results for the associated classes involving many of the new and well-known consequences. We notice that there is symmetry in the results obtained for the new subclasses ${\mathfrak{R}}_{\mathsf{\Sigma },b,c}^{\mu ,\alpha }\left(\lambda ,\delta ,\tau ,\mathsf{\Phi }\right)$ and ${\mathfrak{K}}_{\mathsf{\Sigma },b,c}^{\mu ,\alpha }\left(\lambda ,\delta ,\eta ,\mathsf{\Phi }\right)$, as there is a symmetry for the estimations of the coefficients $a_2$ and $a_3$ for all the subclasses defind in our this paper.

There is no doubt that convex and non-convex functions have a significant impact on optimization. Due to its behavior, convexity also plays a crucial role in the discussion of inequalities. The principles of convexity and symmetry go hand-in-hand. With a growing connection between the two in recent years, we can learn from one and apply it to the other. There have been significant studies on the generalization of Godunova–Levin interval-valued functions in the last few decades, as it has tremendous applications in both pure and applied mathematics. In this paper, we introduce the notion of interval- valued harmonical (h₁, h₂)-Godunova–Levin functions. Using the new concept, we establish a new interval Hermite–Hadamard and Jensen-type inequalities that generalize the ones that exist in the literature. Additionally, we provide some examples to prove the validity of our main results.

The present paper deals with notions from the field of complex analysis which have been adapted to fuzzy sets theory, namely, the part dealing with geometric function theory. Several fuzzy differential subordinations are established regarding the operator $L_{\alpha }^m$, given by $L_{\alpha }^m:{\mathcal{A}}_n\to {\mathcal{A}}_n$, $L_{\alpha }^{m}f\left(z\right)=(1-\alpha )R^{m}f\left(z\right)+\alpha S^{m}f\left(z\right),$ where ${\mathcal{A}}_n=\{f\in \mathcal{H}\left(U\right),f\left(z\right)=z+a_{n+1}{z}^{n+1}+\dots ,z\in U\}$ is the subclass of normalized holomorphic functions and the operators $R^{m}f\left(z\right)$ and $S^{m}f\left(z\right)$ are Ruscheweyh and Sălăgean differential operator, respectively. Using the operator $L_{\alpha }^m$, a certain fuzzy class of analytic functions denoted by $S{L}_{\mathcal{F}}^m\left(\delta ,\alpha \right)$ is defined in the open unit disc. Interesting results related to this class are obtained using the concept of fuzzy differential subordination. Examples are also given for pointing out applications of the theoretical results contained in the original theorems and corollaries.

In this study, we present a numerical method named the logarithmic non-polynomial spline method. This method combines conformable derivative, finite difference, and non-polynomial spline techniques to solve the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. The developed numerical scheme is characterized by a sixth-order convergence and conditional stability. The accuracy of the method is demonstrated with 3D mesh plots, while the effects of time and fractional order are shown in 2D plots. Comparative evaluations with the cubic B-spline collocation method are provided. To illustrate the suitability and effectiveness of the proposed method, two examples are tested, with the results are evaluated using $L_2$ and $L_{\infty }$ norms.