Abstract
In the present study we provide a unified treatment of fractal Hilbert-type inequalities. Our main result is a pair of equivalent fractal Hilbert-type inequalities including a general kernel and weight functions. A particular emphasis is devoted to a class of homogeneous kernels. In addition, we impose appropriate conditions for which the constants appearing on the right-hand sides of the established inequalities are the best possible. As an application, our results are compared with some previously known ones from the literature.
Highlights
The celebrated Hilbert inequality in its integral form asserts that ∞ 0∞ 0 f (x)g(y) x+y dx dy ≤ π sin π p ∞1 p f p(x) dx 1 q gq(y) dy, (1)where f, g : (0, ∞) → R are non-negative integrable functions and p, q is a pair of nonnegative conjugate exponents, i.e., 1 p +
A particular emphasis is devoted to a class of homogeneous kernels
3 Main results we develop a unified treatment of fractal Hilbert-type inequalities
Summary
Where f , g : (0, ∞) → R are non-negative integrable functions and p, q is a pair of nonnegative conjugate exponents, i.e.,. 3, we derive our main result, i.e., a pair of equivalent fractal Hilbert-type inequalities with a general kernel and weight functions. We impose conditions for which the constants appearing on the right-hand sides of the corresponding Hilbert-type inequalities are the best possible. 4 we discuss some particular choices of homogeneous kernels and power weight functions In such a way, we show that the fractal Hilbert-type inequalities presented in this introduction are consequences of our general results. It is not hard to see that our Theorem 1 covers fractal Hilbert-type inequalities (3) and (4) presented in the introduction This follows by choosing a suitable power functions φ, ψ appearing in relations (10) and (11). Cα(0, ∞) is a non-negative homogeneous function of degree –αλ, λ > 0, the following inequalities hold: 0I∞α 0I∞α K (x, y)f (x)g(y).
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