A study of sharp coefficient bounds for a new subfamily of starlike functions

In this article, by employing the hyperbolic tangent function tanhz, a subfamily $\mathcal{S}_{\tanh }^{\ast }$ S tanh ∗ of starlike functions in the open unit disk $\mathbb{D}\subset \mathbb{C}$ D ⊂ C : $$\begin{aligned} \mathbb{D}= \bigl\{ z:z\in \mathbb{C} \text{ and } \vert z \vert < 1 \bigr\} \end{aligned}$$ D = { z : z ∈ C and | z | < 1 } is introduced and investigated. The main contribution of this article includes derivations of sharp inequalities involving the Taylor–Maclaurin coefficients for functions belonging to the class $\mathcal{S}_{\tanh }^{\ast } $ S tanh ∗ of starlike functions in $\mathbb{D}$ D . In particular, the bounds of the first three Taylor–Maclaurin coefficients, the estimates of the Fekete–Szegö type functionals, and the estimates of the second- and third-order Hankel determinants are the main problems that are proposed to be studied here.

Error modeling and singularity analysis of planar parallel mechanisms using optimized expression of joint clearances and link deformation

A fast error modeling method is proposed to analyze the pose deviation of 3-RPR planar parallel mechanisms with multiple revolute joint clearances. The pose error arisen from clearances and limb deformation is modeled from the point of inverse kinematics and estimated numerically. Manipulator poses are represented by elements of the Lie group SE(2) and the discontinuous deformation of limbs due to clearances are modeled by the Heaviside function, which is then approximated with the hyperbolic tangent function. After establishing the error model, an efficient error compensation method is introduced. The proposed error modeling and accuracy analysis method are validated by case study of a 3-RPR mechanism. The results show that the hyperbolic tangent function runs faster than the step function and it provides better convergence. Finally, the singularity of the 3-RPR mechanism with load is analyzed and the results reveal the effects of loads and clearances on mechanism singularities.

Coexisting behaviors of a fraction-order novel hyperbolic-type memristor Hopfield neuron network based on three neurons

The activation function of human neurons is usually regarded as a monotonically differentiable function with upper and lower bounds. Considering the mathematical properties of the hyperbolic tangent function, the activation function can be simulated by a hyperbolic tangent function. In this paper, a fraction-order novel hyperbolic-type memristor Hopfield neuron network (FHMHNN) based on three neurons is proposed, which is achieved using a hyperbolic-type memristor synapse-coupled weight to substitute a coupling-connection weight. The equilibrium points and stability analysis of the FHMHNN are discussed in detail, and the types of generating attractor are determined. Furthermore, the coexisting behaviors of the FHMHNN are described by bifurcation diagram, phase diagram and time diagram. Numerical results show that the FHMHN presents complex dynamical transition, evolving from periodic to chaotic and finally to a stable point with the changes of the memristor coupling weight and inner parameter of the hyperbolic-type memristor. It should be emphasized that the coexisting oscillation behaviors of the FHMHNN under different initial conditions will appear for different inner parameters of the memristor. Theoretical analysis and numerical simulation are basically consistent, revealing that the FHMHNN has the globally coexisting behavior of the asymmetric attractors.