Abstract
A new Hilbert-type linear operator with a composite kernel function is built. As the applications, two new more accurate operator inequalities and their equivalent forms are deduced. The constant factors in these inequalities are proved to be the best possible.
Highlights
In 1908, Weyl 1 published the well-known Hilbert’s inequality as follows: if an, bn ≥ 0 are real sequences, 0 < ∞ n a2n < ∞ and bn2 ∞, ∞ ∞ ambn n 1 m 1m n π ∞∞ a2n bn[2 ] n1 n11.1 where the constant factor π is the best possible
T . a p,φ is the norm of the sequence a with a weight function φ
By setting two monotonic increasing functions u x and v x, a new Hilbert-type inequality, which is with a composite kernel function K u x, v y, and its equivalent are built in this paper
Summary
In 1908, Weyl 1 published the well-known Hilbert’s inequality as follows: if an, bn ≥ 0 are real sequences, 0
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