摘要
设λ,μ≥0,λ+μ=1,则当且仅当p>2时Ax,y∈L^p,x≠y有‖λx+μy‖~2+(p-1)λμ‖x-y‖~2>λ‖x‖~2+μ‖y‖~2。本文简化了这个不等式的证明,把它推广到复L^p空间,并给出了与它等价的几个不等式,它们都是与Clarkson不等式等价的,同时,还推广了反映L^p空间的“2凸性”与“2凹性”的不等式。
Let λ, μ≥0, λ+μ=1, the following inequality is proved by Xu Zongben:if and only if p>2, ‖λx+μy‖~2+(p-1)λμ‖ x-y‖~2>λ‖x‖~2+μ‖y‖~2 for all x, y∈L^p and x≠y. A simple proof of this inequality is given and it is genaralized to complex L^p space. Some inequlities which are equavalent to Clarkson inequality are also given. The inequalities which are known as the '2-convexity' and '2-concavity' of L^p are generalized.
