ABSTRACT
This chapter presents an inequality, which provides a non-trivial estimate on the difference of two sides in a triangle of a Teichmuller space and is stronger than the common triangle inequality of Teichmuller’s distance. It is well-known that the Main Inequality of E. Reich and K. Strebel plays an important role in researching of the extremal problem of quasicon-formal mappings and the theory of Teichmuller spaces. It turns out that the Main Inequality for the compact Riemann surface is a special case of our inequality. The chapter provides a proof of the main inequality for compact Riemann surfaces. The simple quadratic differential (quadratic differential with a single cylinder) is important in the discussion.
