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Abstract: The lemniscatic functions and their inverses play a vital role in various fields of applied science. In classical mechanics, they appear in the exact solutions of the nonlinear pendulum equation and describe the trajectories of particles in central force fields. In electromagnetism, they are involved in the modeling of electric and magnetic field distributions with elliptic symmetry. Moreover, in engineering mathematics and optical system design, lemniscatic functions arise in problems involving wave propagation, signal transmission, and conformal mappings. Their mathematical properties make them valuable tools in both theoretical analysis and computational modeling. Lemniscatic functions, such as Gauss’ arc lemniscate sine and the hyperbolic arc lemniscate sine, arise from the incompletely symmetric elliptic integral of the first kind. In 2007, Neuman introduced additional lemniscatic functions including the arc lemniscate tangent and its hyperbolic counterpart. These functions have attracted considerable attention due to their deep connections with elliptic integrals and their applications in inequality theory. In this paper, we build upon recent work by Wei, He, and Wang (2020), who established Shafer–Fink type inequalities for these lemniscatic functions. Employing methods from real analysis, we present new and sharper refinements of these inequalities. Specifically, we establish double inequalities for Gauss’ arc lemniscate sine, the hyperbolic arc lemniscate sine, arc lemniscate tangent and its hyperbolic counterpart bounded by rational expressions involving radical terms. These results not only improve the existing Shafer–Fink type inequalities but also provide further insight into the monotonicity and convexity properties of the involved lemniscatic functions. Our approach offers a unified framework for deriving tight bounds for all four Neuman lemniscatic functions, contributing to the broader understanding of special function inequalities. These results represent significant refinements over previous known estimates and open avenues for further generalizations.Abstract: The lemniscatic functions and their inverses play a vital role in various fields of applied science. In classical mechanics, they appear in the exact solutions of the nonlinear pendulum equation and describe the trajectories of particles in central force fields. In electromagnetism, they are involved in the modeling of electric and magnetic field d...Learn More
Abstract: This paper is devoted to establishing the best possible upper and lower bounds for the Sándor mean and the identric mean in terms of the harmonic and arithmetic means. These two means, which are closely related to other classical means such as the logarithmic and Seiffert means, have attracted considerable attention in recent studies due to their rich mathematical structure and applications. The primary objective of this article is to refine existing inequalities by determining the sharpest constants in certain double inequalities involving convex combinations of the harmonic and arithmetic means and weighted geometric expressions. To achieve this, we apply a combination of classical inequality techniques and rigorous analysis of the monotonicity properties of auxiliary functions. Specifically, we construct appropriate comparison functions and examine their first and second derivatives to establish strict monotonicity and convexity. These analytic tools allow us to precisely identify the optimal parameters in the refined bounds for the Sándor and identric means. In particular, we show that the best possible bounds for these means can be expressed as weighted combinations of the expressions involving convex combinations of the harmonic and arithmetic means and weighted geometric expressions, where the weights are explicitly determined through monotonicity arguments. As an application of the main results, we derive new and sharp bounds for the functions of the inverse sine and the inverse hyperbolic tangent, which are important in various areas of mathematical analysis and approximation theory. Our findings improve upon several known results in the literature and provide a unified framework for analyzing the relationships among classical and non-classical means.Abstract: This paper is devoted to establishing the best possible upper and lower bounds for the Sándor mean and the identric mean in terms of the harmonic and arithmetic means. These two means, which are closely related to other classical means such as the logarithmic and Seiffert means, have attracted considerable attention in recent studies due to their r...Learn More
Abstract: In this paper, we consider the concept of total derivatives of functions and its relationship with continuity, differentiability, and other concepts. The results indicate that: first, the existence of the total derivative of a bivariate function is a sufficient but not necessary condition for the function's continuity; second, in general, there is no mutually implicative relationship between the existence of the total derivative and the existence of partial derivatives for bivariate functions. However, if the total derivative of a bivariate function equals zero, then the partial derivatives with respect to both independent variables exist and are also zero; third, generally, there is no implicative relationship between the existence of the total derivative and differentiability for bivariate functions. Nevertheless, the total derivative being equal to zero is equivalent to the function being differentiable at that point and the total differential also being equal to zero; fourth, the existence of the total derivative of a bivariate function is a sufficient but not necessary condition for the existence of the directional derivative along any direction at that point. Finally, we presents an application of total derivatives——the mean value theorem, enriching the theory of differential calculus of multivariate functions.Abstract: In this paper, we consider the concept of total derivatives of functions and its relationship with continuity, differentiability, and other concepts. The results indicate that: first, the existence of the total derivative of a bivariate function is a sufficient but not necessary condition for the function's continuity; second, in general, there is ...Learn More
Abstract: Changing the initial state of a population has been proven to have profound impacts on the dynamics of the system and the long-term sustainability of species. Relevant studies have revealed the sensitivity of system parameters, providing a new perspective on system dynamics. The steady-state probability distribution function, mean first passage time and escape rate are obtained by theoretical calculation to discuss the effects of system parameters on the statistical properties of the improved predator-prey model. Obtained results show that the peak of the steady-state probability distribution function decreases with the increase of the multiplicative noise intensity, and its peak decreases with the increase of the birth rate related parameter, and the peak gradually shifts to the right. However, the peak of steady-state probability distribution function gradually becomes larger and more obvious when the correlation strength between noise increases. The mean first passage time T+(x-→xu) and T-(x+→xu) both show a trend of increasing first and then decreasing with the increase of additive noise intensity, but T-(x+→xu) changes faster. With the increase of multiplicative noise intensity, T+(x-→xu) shows a non-monotonic behavior of rapid rise and slow decline, while T-(x+→xu) shows a trend of rising first and then declining, with roughly the same speed of rising and falling. The escape rate W(x±→xu) shows a monotonically decreasing trend with the noise correlation strength, while its function image with the noise intensity shows a monotonically increasing trend. Above results can provide some theoretical guidance for the protection of endangered species from the perspective of statistical physics to ensure that biodiversity conservation is carried out more effectively.Abstract: Changing the initial state of a population has been proven to have profound impacts on the dynamics of the system and the long-term sustainability of species. Relevant studies have revealed the sensitivity of system parameters, providing a new perspective on system dynamics. The steady-state probability distribution function, mean first passage tim...Learn More
