In this paper, we will show some improvements of Heron mean and the refinements of Young’s inequalities for operators and matrices with a different method based on others’ results.
We report on the fabrication of porous ZnO microspheres (pZnO MSs) grafted with thermo-responsive polymers of poly(N-isopropylacrylamide) (PNIPAM) via surface-initiated atom transfer radical polymerization (SI-ATRP) for photocatalysis applications. Photodegradation of Rhodamine B (Rh-B) is used to evaluate the photocatalysis performance of pZnO MSs grafted with PNIPAM (pZnO MSs-PNIPAM). The results show that pZnO MSs-PNIPAM exhibits a reversible temperaturecontrolled photocatalysis activity and good recycling perform-ance. By adjusting the environmental temperature, the photocatalysis property of pZnO MSs-PNIPAM can be turned on/off, resulting in a temperature-controlled switching function of the obvious/unobvious photocatalysis performance below/upon the lower critical solution temperature (LCST) of PNIPAM. Additionally, the dispersion property of pZnO MSs-PNIPAM in water can also be regulated by changing the environment temperature, demonstrating the potential applications in that of photocatalysis-related fields for the graft systems.
Recently some Young type inequalities with Kantorovich constant for square form have been promoted. The purpose of this paper is to give further refinements to them. By using these scalar inequalities, we also obtain some results for operator and matrix.
In this paper, we present some refinements of reverse Young?s inequalities.
Among other results, a refinement of reverse operator Young inequalities
says A?vB + 2?(A?B ? A?B) ? m??M m??M / A?vB, where 0 < mI ? A, B ? MI, ? =
min{v, 1 ? v} and v ? [0, 1], extending a key result in [J. Math. Anal.
Appl. 465 (2018) 267-280] and [Linear Multilinear Algebra 67 (2019)
1567-1578]. Furthermore, we give a reverse of Young?s inequalities due to
[Math. Slovaca 70 (2020), 453-466]. Moreover, we give a generalization of
reverse Young-type inequality, and we also show a new Young-type inequality
which is either better or not uniformly better than the main results in
[Rocky Mountain J. Math. 46 (2016), 1089-1105]. As applications of these
results, we obtain some inequalities for operators, Hilbert-Schmidt norms,
unitarily invariant norms and determinants.
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