在這篇論文中，我們對於一個在Banach空間中的可積分向量值函數，證明了此函數具有 sharp triangle inequality 並且也證明了其三角不等式的reverse inequality， 這之中亦涵蓋了對於 n 個元素的特例，此特例是由 Kato 等學者所發表過的一個結果。此外我們也對一個在LP空間中的向量值函數,推廣了它另一種形式的三角不等式，我們的結果包含了對於兩個元素的特例。另外關於一個改良過的Jensen’s inequality 我們亦討論了其一些相關的性質。

In this thesis, we prove a sharp triangle inequality and its reverse inequality for strongly integrable functions with values in a Banach space X. This contains as a special case a recent result of Kato et al on sharp triangle inequality for n elements. We also discuss a generalized triangle inequality for Lp functions with values in X. It contains as a special case the triangle inequality of the second kind for two elements, which is implied by the Euler-Lagrange type identity. Besides, some properties related to a refined Jensen’s inequality are observed.

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