00:01
Okay, let's first consider the first part of our question.
00:06
Suppose f is an analytic function on a disk, i mean on a disk which is centered at some a, and with radius r.
00:35
We want to show n, given the fact that f of c is less or equal to f on the boundary.
00:49
We can see where c is the boundary.
01:01
C is the boundary for this disk, that means this is actually a circle centered at a, and with radius r.
01:10
Okay, given these conditions, we want to show the derivative of f at a has some relationship with n, and this f, that means the absolute value of the modulus of f and a will be less or equal to the factorial of n times m over r to the power n.
01:46
Okay, when n is equal to zero, let's consider this very simple, that means we want to show f of a, the value of f, will be less or equal to m over r to the power n, which is 1.
01:58
So, let's begin with the discussion for m being zero.
02:07
By all of our assumptions, we know f is analytic on this disk centered at a, then we can use the cauchy integral formula.
02:26
At this point a, i mean f of a by the cauchy integral formula will be 1 over 2 pi i times this integral.
02:34
On the boundary, for the numerator we have f of c, for the denominator we have c minus this specific point a, dz.
02:49
Okay, this is the cauchy integral formula.
02:53
To consider its modulus, we can take the absolute value on both sides.
02:57
On the left side, we have the absolute value of f of a, will be equal to the modulus for 1 over 2 pi i will be 1 over 2 pi.
03:17
For convenience, let's define this function as f of c.
03:21
We know f of c is again an analytic function by our definition.
03:28
So, this will be less or equal to 1 over 2 pi.
03:32
Use the inequality between absolute value of integral and the integral of absolute value.
03:44
We know it will be less or equal to the absolute value of this guy.
03:50
However, if we consider the absolute value of f of c, we know on the boundary, absolute value of z.
04:04
As z is on the boundary, so the denominator will be just equal to r.
04:09
For the numerator, it will be less or equal to m.
04:12
So, this will be m times r dz.
04:22
Again, z is on the circle, so the integral for the circle over a constant function will be just equal to 1 over 2 pi times m over r times the arc length of the circle, which is 2 pi times r.
04:43
Do some cancellation.
04:45
We know f of c will be less or equal to m, which just matches our expression here.
04:57
Okay, so this just gives us a hint of doing things like that.
05:02
For other derivative, we want to use the so -called cauchy, generalized cauchy, generalized cauchy integral formula...