Four-point distortion theorem for complex polynomials. (English) Zbl 1300.30009
In the paper a result in connection with the distortion of the cross ratio of four points under the mapping defined by a complex polynomial with restricted critical values is proved. The corollaries deduced from this result include some inequalities which involve the absolute value and certain coefficients of a polynomial. In particular, the author obtains an exact lower bound for the maximum moduli of the critical values of a polynomial \(P\) of degree \(n\) which is normalized by the following conditions: i) the value of \(P\) at the origin is zero and ii) its derivative at the origin is non-zero.
Reviewer: Sanjib Kumar Datta (Kalyani)
MSC:
30C10 | Polynomials and rational functions of one complex variable |
30C35 | General theory of conformal mappings |
30C85 | Capacity and harmonic measure in the complex plane |
Keywords:
Chebyshev polynomial; critical values; distortion theorems; modulus of a doubly connected domain; cross ratio of four pointsReferences:
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