Multidimensional Hilbert-Type Inequalities Obtained via Local Fractional Calculus
- PDF / 767,048 Bytes
- 14 Pages / 439.37 x 666.142 pts Page_size
- 109 Downloads / 218 Views
Multidimensional Hilbert-Type Inequalities Obtained via Local Fractional Calculus Mario Krni´c1 · Predrag Vukovi´c2
Received: 21 July 2019 / Accepted: 31 January 2020 © Springer Nature B.V. 2020
Abstract In this paper we give a unified treatment of multidimensional fractal Hilbert-type inequalities. More precisely, we establish a Hilbert-type inequality with a general local fractional continuous kernel and weight functions. A particular emphasis is dedicated to a class of inequalities with a homogeneous kernel. Namely, we impose some weak conditions for which the constants appearing on the right-hand sides of such Hilbert-type inequalities are the best possible. As an application, we discuss some particular choices of homogeneous kernels and power weight functions. Mathematics Subject Classification 26D15 Keywords Hilbert inequality · Conjugate parameters · Homogeneous function · Local fractional calculus · Multidimensional form
1 Introduction Let p and q be a pair of non-negative conjugate parameters i.e. p1 + q1 = 1, p > 1. The famous Hilbert integral inequality (see [8]) in its simplest form asserts that π f (x)g(y) dxdy ≤ f p gq , (1) x +y sin πp R2+ where f ∈ Lp (R+ ) and g ∈ Lq (R+ ) are non-negative functions. The constant sinπ π , appearp ing on the right-hand side of (1), is the best possible in the sense that it can not be replaced by a smaller positive constant so that the inequality still holds.
B M. Krni´c
[email protected] P. Vukovi´c [email protected]
1
Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia
2
Faculty of Teacher Education, University of Zagreb, Savska cesta 77, 10000 Zagreb, Croatia
M. Krni´c, P. Vukovi´c
After discovering, the Hilbert inequality was extensively studied by numerous authors. A rich collection of generalizations included inequalities with more general kernels, weight functions and integration domains, extension to a multidimensional case, as well as refinements of the initial Hilbert inequality. Although classical, the Hilbert inequality is still topic of interest to numerous authors. For a detailed review of the starting development of the Hilbert inequality the reader is referred to [8], while some recent results are collected in [10]. On the other hand, an interesting subject in connection to classical inequalities is their extension to fractal spaces via the local fractional calculus. Primary task of the local fractional calculus is to handle various non-differentiable problems appearing in complex systems of the real-world phenomena. In particular, the non-differentiability occurring in science and engineering has been modeled by the local fractional ordinary or partial differential equations. Although arising from real-world phenomena, the local fractional calculus is also an important tool in pure mathematics. Recently, a whole variety of classical real inequalities has been extended to hold on certain fractal spaces. For the reader’s convenience, let (α) (α) (α) a Ib f (x) and a Ib [a Ib h(x
Data Loading...
