# Question related to conformal mappings

• sari
In summary, to prove that |a_n| < |a_1|, we used the Cauchy inequality and the fact that P(z) is 1-1 on the open unit disk D. We showed that the maximum value of P(z) on the unit circle must be less than the maximum value of P(z) on the unit disk, and therefore, |a_n| < |a_1| for all n.
sari
1. P(z) = a_0 + a_1*z + a_2*z^2 + ... + a_n*z^n is 1-1 on the open unit disk D = {z: |z|<1}. Prove that |a_n| < |a_1|\n.

2.

3. P(z) is analytic and 1-1, and therefore conformal, so |P'(z)|>0 for every z in D.
So 0 < |P'(z)|=|a_1 + 2a_2*z +...+ na_n*z^n-1| < |a_1| + ...+|na_n|.

Also the inverse function P^-1(z) is conformal in P(D) and |P^-1(z)|<1.
We also know that the derivative d\dz P^-1(z) = 1\P'(P^-1(z)) and that |d\dz P^-1(z)|>0.

Additionally, we know that a_k = (P^(k) (0)\ k!) .

The only other things I thought of that could be useful are the Cauchy inequality |a_k| < M(r)\ r^k, where M(r) = max{f(z): |z| = r }, and maybe Shwarz's Lemma, though since P(z) doesn't map 0 to 0 and doesn't map the unit disc to itself, I don't see how it would be relevant.

Would appreciate any help very much ... I don't really know where to go from here.

To prove that |a_n| < |a_1|, we can use the Cauchy inequality and the fact that P(z) is 1-1 on the open unit disk D.

From the Cauchy inequality, we know that |a_n| < M(1)\ 1^n = M(1) for all n, where M(1) is the maximum value of P(z) on the unit circle.

Since P(z) is 1-1 on D, it must map each point on the unit circle to a unique point inside the unit disk. Therefore, the maximum value of P(z) on the unit circle must be less than the maximum value of P(z) on the unit disk.

So, we have |a_n| < M(1) < M(r) for all n and for all r<1.

Using the Cauchy inequality again, we have |a_1| < M(r)\ r for all r<1.

Since |a_1| < M(r)\ r for all r<1 and |a_n| < M(r) for all n and for all r<1, we can conclude that |a_n| < |a_1| for all n.

Therefore, we have proven that |a_n| < |a_1| for all n, as required.

## 1. What is a conformal mapping?

A conformal mapping is a type of mathematical transformation that preserves angles between intersecting curves. In simpler terms, it is a function that preserves the shape and angles of a figure when it is mapped onto another figure.

## 2. How is a conformal mapping different from other types of mappings?

Unlike other types of mappings, a conformal mapping preserves the angles between intersecting curves. This means that the local shape and orientation of the curves are maintained, making it useful for applications such as cartography and fluid dynamics.

## 3. What are some common examples of conformal mappings?

Some common examples of conformal mappings include the Mercator projection used in maps, the stereographic projection used in astronomy, and the complex exponential function used in mathematics.

## 4. What are the properties of conformal mappings?

Conformal mappings have several key properties, including preserving angles, conformality, and bijectivity. They also have a conformal factor, which is a function that describes the scale and orientation changes in the mapping.

## 5. What are some real-life applications of conformal mappings?

Conformal mappings have many practical applications, including in cartography, where they are used to create accurate maps; in fluid dynamics, where they are used to model the flow of fluids; and in computer graphics, where they are used to create realistic 3D models.

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