In the article, we establish several new inequalities for the generalized trigonometric and hyperbolic functions with one parameter, generalize the well known Mitrinović-Adamović, Lazarević, Huygens-type, Wilker-type and Cusa-Huygens-type inequalities to the cases of the generalized trigonometric and hyperbolic functions with one parameter.

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Inequalities for generalized trigonometric and hyperbolic functions with one parameter

Abstract In the article, we present the best possible parameters α1, β1, α2, β2 ∈ ℝ and α3, β3 ∈ [1/2, 1] such that the double inequalities α1C(a,b)+(1−α1)A(a,b)<T3(a,b)<β1C(a,b)+(1−β1)A(a,b),α2C(a,b)+(1−α2)Q(a,b)<T3(a,b)<β2C(a,b)+(1−β2)Q(a,b),C(α3;a,b)<T3(a,b)<C(β3;a,b) $$\\begin{array}{} \\begin{split} \\displaystyle \\alpha_{1}C(a, b)+(1-\\alpha_{1})A(a, b) & \\lt T_{3}(a, b) \\lt \\beta_{1}C(a, b)+(1-\\beta_{1})A(a, b), \\\\ \\alpha_{2}C(a, b)+(1-\\alpha_{2})Q(a, b) & \\lt T_{3}(a, b) \\lt \\beta_{2}C(a, b)+(1-\\beta_{2})Q(a, b), \\\\ C(\\alpha_{3}; a, b) & \\lt T_{3}(a, b) \\lt C(\\beta_{3}; a, b) \\end{split} \\end{array}$$ hold for a, b > 0 with a ≠ b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean, Q(a,b)=a2+b2/2 $\\begin{array}{} \\displaystyle Q(a, b)=\\sqrt{\\left(a^{2}+b^{2}\\right)/2} \\end{array}$ is the quadratic mean, C(a, b) = (a2 + b2)/(a + b) is the contra-harmonic mean, C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] is the one-parameter contra-harmonic mean and T3(a,b)=(2π∫0π/2a3cos2⁡θ+b3sin2⁡θdθ)2/3 $\\begin{array}{} T_{3}(a,b)=\\Big(\\frac{2}{\\pi}\\int\\limits_{0}^{\\pi/2}\\sqrt{a^{3}\\cos^{2}\\theta+b^{3}\\sin^{2}\\theta}\\text{d}\\theta\\Big)^{2/3} \\end{array}$ is the Toader mean of order 3.

Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means

This sequel is concerned with the analysis of projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks (FOMBNNs) with mixed time delays via hybrid controller. Firstly, a new type of hybrid control scheme, which is the combination of open loop control and adaptive state feedback control is designed to guarantee the global projective lag synchronization of the addressed FOMBNNs model. Secondly, by using a Lyapunov–Krasovskii functional and Barbalet’s lemma, a new brand of sufficient criterion is proposed to ensure the projective lag synchronization of the FOMBNNs model considered. Moreover, as special cases by using a hybrid control scheme, some sufficient conditions are derived to ensure the global projective synchronization, global complete synchronization and global anti-synchronization for the FOMBNNs model considered. Finally, numerical simulations are provided to check the accuracy and validity of our obtained synchronization results.

https://www.mdpi.com/2227-7390/7/8/759/pdf

Hybrid Control Scheme for Projective Lag Synchronization of Riemann–Liouville Sense Fractional Order Memristive BAM NeuralNetworks with Mixed Delays

Asymptotically stable high-order neutral cellular neural networks with proportional delays and D operators

In the article, we present several Hermite–Hadamard type inequalities for the co-ordinated convex and quasi-convex functions and give an application to the product of the moment of two continuous and independent random variables. Our results are generalizations of some earlier results. Additionally, an illustrative example on the probability distribution is given to support our results.

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Hermite–Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications

Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle–focus type

This paper presents a Filippov plant disease model incorporating an economic threshold of infected-susceptible ratio, above which control strategies of replanting or removing are needed to be carried out. Based on the Filippov approach, we study the sliding mode dynamics and further the global dynamics. It is shown that there is a unique equilibrium, which is a disease-free equilibrium, an endemic equilibrium or a pseudo-equilibrium. Moreover, the equilibrium is proved to be globally asymptotically stable. Our results indicate that the control goal can be achieved by taking appropriate replanting and removing rate.

Global dynamics of a filippov plant disease model with an economic threshold of infected-susceptible ratio

Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure

Based on the integrated pest management strategy, we propose and analyze a Filippov predator-prey model. Different from existing works, the feature of the model is that two thresholds are involved, one ( x T ) for pest and the other ( y T ) for natural enemy. We distinguish three cases according to the value of x T . Then we classify the dynamics for each case considering the relationship of the levels of the natural enemy at equilibria to y T by applying Filippov theory. Specifically, we investigate the sliding mode dynamics and global dynamics. It is shown that, for different parameter values, the proposed system can admit coexistence of multi-attractors including a real equilibrium, a pseudo-equilibrium, and a pseudo-attractor ( x T , y T ) . These theoretical results are also illustrated with numerical simulations. Our findings indicate that proper control strategies can prevent the outbreak of pest disaster in ecological agriculture. 

Global dynamics of a filippov predator-prey model with two thresholds for integrated pest management

Global dynamics of a Filippov susceptible, infectious, quarantine, and recovered model subject to delayed relay control are investigated in this work. The global stability of equilibria is obtained by means of the generalized Lyapunov method. In addition, combining the Poincaré maps, the displacement function, and a Melnikov-like method, we derive some sufficient conditions to ensure the existence, uniqueness, and global stability of a slowly oscillating periodic solution. Numerical simulations are performed to explore the effectiveness of the quarantine control measure and the effects of the delay. 

Jiafu Wang

Papers

Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle–focus type

Global dynamics of a filippov plant disease model with an economic threshold of infected-susceptible ratio

Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure

Global dynamics of a filippov predator-prey model with two thresholds for integrated pest management

Global Dynamics of a Filippov SIQR Model with Delayed Relay Control