- #1
smyroosh
- 2
- 0
How to prove the following inequality: for complex z such that Re z < 0 :
[tex]\left| e^z-1\right| < \left| z\right|[/tex] ?
[tex]\left| e^z-1\right| < \left| z\right|[/tex] ?
A complex inequality is an inequality that involves complex numbers. It is similar to a regular inequality, but instead of just using real numbers, it also includes imaginary numbers.
To prove a complex inequality, you need to show that it is true for all possible complex numbers. This can be done by using algebraic manipulations and properties of complex numbers, such as the triangle inequality and the modulus inequality.
"Re z" refers to the real part of the complex number z. In other words, it is the part of z that does not involve the imaginary unit, i. Therefore, the inequality "Re z < 0" means that the real part of z is less than 0.
One example of a complex inequality is "|z + 2i| < |z + 3i|". This means that the distance between the complex number z and 2i is less than the distance between z and 3i.
Proving a complex inequality is important because it allows us to determine the values of complex numbers that satisfy the inequality. This can be useful in various areas of mathematics, such as complex analysis and differential equations.