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:
Application of basic inequality of extended data;
1. When solving problems with basic inequalities, we must pay attention to the preconditions of application: "one positive", "two definite" and "three phases". The so-called "one positive" is a positive number, "two definite" means that the sum or product is constant when the basic inequality is applied to find the maximum value, and "three phases are equal" means that the equal sign condition is met.
2. When using the basic inequality to find the maximum value, we should flexibly deform the formula according to the characteristics, complement the product and sum it in the form of a constant, and then use the basic inequality.
3. There are usually two methods to solve the condition maximum:
(1) First, the elimination method, that is, the functional relationship between two quantities is established according to the conditions, and then converted into the maximum value of the function to solve;
(2) The second is the flexible deformation condition. By substituting the constant "1", the formula that the sum or product is constant is constructed, and then the maximum value is solved by using the basic inequality.
Baidu encyclopedia-inequality