Geodesic
Exact inequality between uniform norms of an algebraic polynomial and its real part on concentric circles in the complex plane
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 4, pp. 254-263

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In the class $\mathcal P_n^*$ of algebraic polynomials of a complex variable of degree at most n with complex coefficients and a real constant term, we estimate the uniform norm of a polynomial $P_n\in\mathcal P_n^*$ on the circle $\Gamma_r=\{z\in\mathbb C\colon|z|=r\}$ of radius $r>1$ in terms of the norm of its real part on the unit circle $\Gamma_1$. More precisely, we study the best constant $\mu(r,n)$ in the inequality $\|P_n\|_{C(\Gamma_r)}\leq\mu(r,n)\|\operatorname{Re}P_n\|_{C(\Gamma_1)}$. Necessary and sufficient conditions for the equality $\mu(r,n)=r^n$ are found.
Keywords: inequalities for algebraic polynomials, uniform norm, circle in the complex plane. 
A. V. Parfenenkov. Exact inequality between uniform norms of an algebraic polynomial and its real part on concentric circles in the complex plane. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 4, pp. 254-263. http://geodesic.mathdoc.fr/item/TIMM_2010_16_4_a23/
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