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Note on Bernstein’s inequality for the third derivative of a polynomial. (English) Zbl 1060.30003
Summary: Given a polynomial \(p(z)= \sum^n_{j=0} a_j z^j\), we give the best possible constant \(c_3(n)\) such that \(\| p'''\|+ c_3(n)|a_0|\leq n(n- 1)(n-2)\| p\|\), where \(\|\;\|\) is the maximum norm on the unit circle \(\{z: |z|= 1\}\). Most of the computations are done with a computer.
MSC:
30A10 Inequalities in the complex plane
26D05 Inequalities for trigonometric functions and polynomials
26D10 Inequalities involving derivatives and differential and integral operators
30C10 Polynomials and rational functions of one complex variable
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