On Minkowski ’ s inequality and its application

@inproceedings{Zhao2011OnM,
  title={On Minkowski ’ s inequality and its application},
  author={Chang-Jian Zhao and Wing-Sum Cheung},
  year={2011}
}
with equality if and only if f and g are proportional. For p <0, we assume that f(x), g (x) >0. An (almost) improvement of Minkowski’s inequality, for p Î R\{0}, is obtained in the following Theorem: Theorem 1.2 Let f(x), g(x) ≥ 0 and p >0, or f(x), g(x) >0 and p <0. Let s, t Î R\{0}, and s ≠ t. Then (i) Let p, s, t Î R be different, such that s, t >1 and (s t)/(p t) >1. Then ∫ (f (x)+g(x))pdx ≤ [(∫ f s(x)dx )1/s + (∫ gs(x)dx )1/s]s(p−t)/(s−t) 

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