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. 
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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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