Trigonometric inequality, that is, the sum of two sides in a triangle is greater than the third side, sometimes refers to the formula of trigonometric function connected by an inequality symbol (not introduced here). Trigonometric inequality is simple, but it is the most basic conclusion in plane geometric inequality.

2. Average inequality

Mean inequality, also known as mean inequality and mean inequality, is an important formula in mathematics. The content of the formula is Hn≤Gn≤An≤Qn, that is, the harmonic average does not exceed the geometric average, the geometric average does not exceed the arithmetic average, and the arithmetic average does not exceed the square average.

3. Cauchy inequality

Cauchy inequality was obtained when Cauchy, a great mathematician, studied the problem of "flow number" in mathematical analysis.

But from a historical point of view, this inequality should be called Cauchy-Bunyakovski-Schwartz inequality, because it is the latter two mathematicians who generalize it independently in integral calculus that makes this inequality applied to a nearly perfect degree.

Cauchy inequality is an inequality discovered by Cauchy during his research. It is widely used to solve the related problems of inequality proof, so it is very important in the promotion of higher mathematics and is one of the research contents of higher mathematics.

4. Geometric mean inequality

The root number ab is called geometric average, which embodies a geometric relationship, that is, a vertical line is drawn at any point on the diameter of a circle, and the two parts with separate diameters are A and B, then half the length of the vertical line in the circle is the root number ab, (a+b)/2≥ the root number ab! This is its geometric meaning, which is why it is called geometric average.

Arithmetic-geometric mean inequality, referred to as arithmetic mean inequality, is a common and basic inequality, which is manifested as a constant inequality between arithmetic mean and geometric mean.

5. Young's inequality

Young's inequality, also known as Young's inequality, is a special case of weighted arithmetic-geometric mean inequality, and Young's inequality is a quick way to prove Holder's inequality.