The value range of complex modulus T ∈ R, t ≠ 0, z = t / (1 + T) + I * (1 + T) / T

The value range of complex modulus T ∈ R, t ≠ 0, z = t / (1 + T) + I * (1 + T) / T

│z│=√{[t/(1+t)]²+[(1+t)/t]²}
≥√{2[t/(1+t)]²*[(1+t)/t]²}=√2
Take the equal sign if and only if t = - 1 / 2

How to find the principal value of complex number
It is to find the principal value of the argument angle, such as the principal value of the cube root of 1 + I

Take the real part of the complex number as the x-axis and the imaginary part as the y-axis as the plane rectangular coordinate system, and draw the point corresponding to the complex number, the straight line connecting this point to the origin. The inclination angle of this line is the principal value of the argument, for example, finding the cube root of 1 + I = the principal value of 1-I - π / 4

Let a and B be arbitrary real numbers, and prove that 16x / (x × x + 8) < B × b-3b + 21 / 4

b×b-3b+21/4=(b-3/2)^+3>=3
x=0
0

Let f (x) = 16x / (x ^ 2 + 8x) (x > 0), it is proved that for any real number B, f (x) always exists

f(x)=16x/(x^2+8x)
=16/(x+8)
The function decreases monotonically on (0, + ∞)
The range is (0,2)
b^2-2b+4=(b-1)^2+3≥3＞f(x)