Abstract
In this paper, the modular-type operator norm of the general geometric mean operator over spherical cones is investigated. We give two applications of a new limit process, introduced by the present authors, to the establishment of Polya-Knopp-type inequalities. We not only partially generalize the sufficient parts of Persson-Stepanov’s and Wedestig’s results, but we also provide new proofs to these results.
Highlights
We mean that E = s> sA for some Borel measurable subset A of the unit sphere n
CV +(I) denotes the set of all nonnegative convex functions defined on an open interval I in R, DK is the space of those f such that Kf (x) is well defined for almost all x ∈ E, and Lp (v dx) is the set of all real-valued Borel measurable f with
Applying the Lebesgue dominated convergence theorem again, it follows from Lemma . that lim m→∞
Summary
For the particular case that (s) = |s|, k(x, t) = , there are two other types of estimates We focus on the evaluation of K ∗ for the following case of They obtained the following estimates by means of the formula (GKf )(x) = lim → + [K(f )] / (x):. F+ = { > } and the second inequality in APS(p, q) := sup v(t) –p∗ dt x∈E Sx v(y) –p∗ dy u(t) dt
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