1, trigonometric inequality

Triangle inequality means that the sum of two sides in a triangle is greater than the third side, which is the most basic conclusion in plane geometric inequality. Generalized Ptolemy Theorem, euler theorem and Euler Inequality all use this inequality to derive inequality relations.

2. Average inequality

Hn≤Gn≤An≤Qn is called mean inequality, that is, the harmonic mean does not exceed the geometric mean, the geometric mean does not exceed the arithmetic mean, and the arithmetic mean does not exceed the square mean, which is abbreviated as "adjusting several formulas".

3. Binary mean inequality

Binary mean inequality means that the arithmetic mean of two positive real numbers is greater than or equal to their geometric mean. The formula is: A2+B2 ≥ 2ab; Generally speaking, if A 1, A2, A3, ..., an is a positive real number, there is an average inequality:

4. Young's inequality

Young's inequality is also called Young's inequality. Young's inequality is a special case of weighted arithmetic-geometric mean inequality, and its general form is: assuming that both A and B are non-negative real numbers, p > 1, 1/p+ 1/q= 1, then:

The equal sign holds if and only if a p = b q.

5. 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, and its general form is:

6. Holder inequality

Held inequality is an inequality in mathematical analysis, which is based on Otto H? Lder). This is a basic inequality that reveals the relationship between Lp spaces. Let p > 1, 1/p+ 1/q= 1, let a 1, an and b 1, and bn be nonnegative real numbers, then:

Baidu encyclopedia-inequality