We discuss a class of new nonlinear weakly singular difference inequality, which is solved by change of variable, discrete Hölder inequality, discrete Jensen inequality, the mean-value theorem for integrals and amplification method, and Gamma function. Explicit bound for the unknown function is given clearly. Moreover, an example is presented to show the usefulness of our results.

By means of the weight functions, the idea of introducing parameters and the technique of real analysis related to the beta and gamma functions, a new reverse Hardy–Hilbert-type integral inequality with the homogeneous kernel as $\frac{1}{(x + y)^{\lambda + m + n}}$ 1 ( x + y ) λ + m + n ($\lambda > 0$ λ > 0 ) involving two derivative functions of higher order is given. As applications, the equivalent statements of the best possible constant factor related to several parameters are considered, and some particular inequalities are obtained.

Abstract-In this paper, we investigate a nonlinear weakly singular integral inequality. The upper bounds of the embedded unknown functions are estimated explicitly by the definitions and rules of conformable fractional differential and conformable fractional integration, the techniques of change of variable, and the method of amplification. The derived results can be applied in the study of qualitative properties of solutions of conformable fractional integral equations.

By means of the weight functions, Hermite-Hadamard’s inequality and the technique of real analysis, a new more accurate reverse half-discrete Hilbert-type inequality involving one higher-order derivative function is given. The equivalent conditions of the best possible constant factor related to a few parameters, the equivalent forms, several particular inequalities and the kind of reverses are considered.

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