Geodesic
Some remarks on rotation theorems for complex polynomials
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 369-376

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For any complex polynomial $P(z)=c_0+c_1z+...+c_nz^n, c_n\not=0,$ having all its zeros in the unit disk $|z|\le 1,$ we consider the behavior of the function (arg$P(e^{i\theta}))'_{\theta}$ when the real argument $\theta$ changes. We give some sharp estimates of this function involving of the values of $P(e^{i\theta}),$ arg$P(e^{i\theta})$ or the coefficients $c_k, k=0,1,n-1,n.$
Keywords: complex polynomials, rotation theorems, inequalities, boundary Schwarz lemma, rational functions. 
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     author = {V. N. Dubinin},
     title = {Some remarks on rotation theorems for complex polynomials},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
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     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a44/}
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V. N. Dubinin. Some remarks on rotation theorems for complex polynomials. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 369-376. http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a44/