A Hilbert-type fractal integral inequality and its applications

Abstract

By using thefractal theory and the methods of weight function, a Hilbert-type fractal integral inequality and its equivalent form are given. Their constant factors are proved being the best possible, and their applications are discussed briefly.

A Hilbert-type integral inequality under configuring free power and its applications

“…During decades, inequality (1) has been extensively studied by numerous authors, evolved into a lot of meaningful results, which include the research of parametric quantization, mixed kernels, homogeneous kernels and non-homogeneous kernels, the extensions of fractal space, etc. (see [3][4][5][6][7][8][9][10][11][12][13][14][15]). In 2011, Yang gave an integral inequality of Hilbert type with exponential kernel as follows (see [16]):…”

A unified approach to fractal Hilbert-type inequalities

“…Our next intention is to impose the condition on parameters A 1 and A 2 for which the constants appearing on the right-hand sides of inequalities (13) and (14) are the best possible. It should be noticed here that if…”

“…We will show that if the parameters A 1 and A 2 are related by (20), then the constants appearing on the right-hand sides of (13) and (14) are the best possible. In fact, if (20) holds, inequalities (13) and 14reduce to…”

“…Recently, Liu and Sun [13] established a pair of equivalent fractal Hilbert-type inequalities expressed in terms of the above fractional integrals. Namely, they showed that if 1 p + 1 q = 1, p > 1, 0 < α ≤ 1, and if f , g ∈ C α (a, b) are non-negative functions, then the following inequalities hold:…”

A non-differentiable solution for the local fractional telegraph equation

“…Local fractional calculus [1] has played an important role in the field of mathematical science and mathematical physics, such as the generalized convex [2] and s-convex [3,4] functions on fractal sets, and Pompeiu-type [5], Steffensen [6], Hermite-Hadamard [7], Holder [8], Hilbert [9], Korteweg-de Vries [10], Burgers [11], Boussinesq [12], heat conduction [13], diffusion [14,15], tricomi [16], Goursat [17], and others [18][19][20][21].…”