On an Extended Hardy-Hilbert's Inequality in the Whole Plane

Pillar[5]arene based supramolecular polymer for a singlet oxygen reservoir

“…8,12,13 Recently, a series of organic functional systems with diphenylanthracene backbones have been constructed to implement the storage and controlled release of 1 O 2 . [14][15][16][17][18][19][20] Xiao's group used diphenylanthracene and porphyrin derivatives to prepare a covalent organic polymer which can capture the 1 O 2 produced by the photodynamic process of porphyrins and which can release 1 O 2 after heating to 110 °C. 19 Other polymer materials with diphenylanthracene units connected directly on their side chains through covalent bonds have also been synthesized for self-carrying 1 O 2 by Ling et al 18 However, being based on covalent bonds, the construction of these materials is time-consuming and cost-intensive, which will also limit the selection of diverse functional units that have better capability for 1 O 2 storage and release.…”

On a class of Hilbert-type inequalities in the whole plane involving some classical kernel functions

“…It is well known that the classical Hilbert-type inequalities are usually constructed in the first quadrant. In recent years, with more and more researchers participating in the study of Hilbert-type inequalities, some researchers extended the range of variables to the whole plane and established many valuable new results (see [7,16–18,21,33,37,41]). Inspired by the related results in these literatures, we will construct the following new inequalities: where , , , and .…”