Some New Results Related To Minkowski Inequalities And Their Applications

Minkowski inequality is one of the classical inequality, it has important applications in geometry, optimal control, matrix theory, operator theory, function and so on. Since the discovery of this inequality, there has been no interruption of its research for more hundred years. With the conditions to be strengthened, weakened, or changed, we got different expressions for Minkowski inequality in various mathematical areas and its various improvements, promotions and applications.In this paper, we study the discrete form and Riemann integral form of Minkowski inequalities in one-dimensional space.H(o|¨)lder inequality is the basis tool to study Minkowski inequality. For study Minkowski inequalities, first of all, using H(o|¨)lder inequalities on the basis of [23], creating a new mapping and studying properties of the new mapping and mappings defined in [23], we obtain some new results related to H(o|¨)lder inequalities, to prepare for researching Minkowski inequalities.Second, changing the variable changes in Minkowski inequalities, establishing two mappings, and using new results related to H(o|¨)lder inequalities to study their properties, we obtain some new results related to Minkowski inequalities. In addition, we also use two series and the basic properties of Minkowski inequalities to go further research them, we got some refinements of Minkowski inequalities.Finally, replacing variables in these new results with some specific functions, we get some new results related to the p -logarithmic mean.Results in this paper enriche inequality theory and will play an important role in promoting the developments of other related disciplines.