A class of new trigonometric inequalities and their sharpenings

Some sufficient or necessary conditions for Schur-convexity of a function of two variables F(x, y) = (f (y)f (x))/(g(y)g(x)) were considered. These results are applied to establish a class new inequalities in a triangle. In the fourth section we prove two theorems for a kind of symmetric function. These theorems are used to sharpen some of the inequalities and yield two inequalities in the last section.