Uniform Treatment of Jensen’s Inequality by Montgomery Identity

We generalize Jensen’s integral inequality for real Stieltjes measure by using Montgomery identity under the effect of n − convex functions; also, we give different versions of Jensen’s discrete inequality along with its converses for real weights. As an application, we give generalized variants of Hermite–Hadamard inequality. Montgomery identity has a great importance as many inequalities can be obtained from Montgomery identity in q − calculus and fractional integrals. Also, we give applications in information theory for our obtained results, especially for Zipf and Hybrid Zipf–Mandelbrot entropies.

New discrete inequalities of Hermite–Hadamard type for convex functions

We introduce new time scales on $\mathbb{Z}$ Z . Based on this, we investigate the discrete inequality of Hermite–Hadamard type for discrete convex functions. Finally, we improve our result to investigate the discrete fractional inequality of Hermite–Hadamard type for the discrete convex functions involving the left nabla and right delta fractional sums.

Positive Periodic Solutions of a Periodic Discrete Competitive System Subject to Feedback Controls

Species living in a fluctuating medium and human exploitation activities might result in the duration of continuous changes. Such changes can be well-approximated as feedback controls. In this contribution a periodic discrete competitive system subject to feedback controls is proposed. By using the methods of discrete inequality, fixed point theorem, and analysis techniques, a good understanding of the existence and global asymptotic stability of positive periodic solutions is obtained. Some numerical investigations are provided to verify our analytical results.

A Generalized Nonlinear Sum-Difference Inequality of Product Form

We establish a generalized nonlinear discrete inequality of product form, which includes both nonconstant terms outside the sums and composite functions of nonlinear function and unknown function without assumption of monotonicity. Upper bound estimations of unknown functions are given by technique of change of variable, amplification method, difference and summation, inverse function, and the dialectical relationship between constants and variables. Using our result we can solve both the discrete inequality in Pachpatte (1995). Our result can be used as tools in the study of difference equations of product form.