On the growth of polynomials
Tóm tắt
The Erdős–Lax Theorem states that, if P(z) is a polynomial of degree n having no zeros in
$$|z|<1$$
, then
$$\max_{|z|=1}|P'(z)|\leq \frac{n}{2}\max_{|z|=1}|P(z)|.$$
The problem of generalizing the Erdős–Lax Theorem to the class of polynomials having no zeros in
$$|z|
Tài liệu tham khảo
S. N. Bernstein, Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d’une variable réelle, Gauthier-Villars (Paris, 1926).
R. Dhankhar, N. K. Govil and P. Kumar, On sharpening of inequalities for a class of polynomials satisfying \(p(z)\equiv z^n p(1/z)\) , Studia Sci. Math. Hungar., 57 (2020),255–266.
P. Erdős, On extremal properties of derivatives of polynomials, Ann. Math., 41 (1940),310–313.
E. Frank, On the real parts of the zeros of complex polynomials and applications to continued fraction expansions of analytic functions, Trans. Amer. Math. Soc.,62 (1947), 272–283.
R. B. Gardner, N. K. Govil and G. V. Milovanović, Extremal Problems and Inequalities of Markov–Bernstein Type for Algebraic Polynomial, Academic Press/Elsevier (2022).
N. K. Govil, On the derivative of a polynomial, Proc. Amer. Math. Soc., 41 (1973),543–546.
N. K. Govil and Q. I. Rahman, Functions of exponential type not vanishing in a half-plane and related polynomials, Trans. Amer. Math. Soc., 137 (1969), 501–517.
N. K. Govil, On a theorem of S. Bernstein, Proc. Nat. Acad. Sci. (India), 50A (1980),50–52.
N. K. Govil and P. Kumar, On \(L^p\) inequalities involving polar derivative of a polynomial,Acta. Math. Hungar., 152 (2017), 130–139.
P. Kumar, On the derivative of a polynomial, Proc. Edinburgh Math. Soc., 65 (2022),303–310.
P. Kumar, On the Erdős–Lax inequality, Comptes Rendus Math., 360 (2022), 1081–1085.
P. D. Lax, Proof of a conjecture due to Erdős on the derivative of a polynomial, Bull.Amer. Math. Soc., 50 (1944), 509–513.
M. A. Malik, On the derivative of a polynomial, J. London Math. Soc., 1 (1969),57–60.
M. Marden, Geometry of Polynomials, Math. Surveys, vol. 3, American Mathematical Society, (Providence, RI, 1966).
G. V. Milovanović, D. S. Mitrinović and Th. M. Rassias, Topics in Polynomials:Extremal Properties, Inequalities, Zero, World Scientific Publishing Co. (Singapore,1994).
A. Mir, A note on an inequality of Paul Turán concerning polynomials, Ramanujan J.,56 (2021), 1061–1071.
G. Pólya and G. Szegő, Aufgaben und Lehrsätze aus der Analysis, Springer (Berlin,1925).
Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press (2002).
E. B. Saff and T. Sheil-Small, Coefficient and integral mean estimates for algebraic and trigonometric polynomials with restricted zeros, J. London Math. Soc., 9 (1974), 16–22.
N. A. Rather, A. Bhat and M. Shafi, Integral inequalities for the growth and higher derivative of polynomials, J. Contemp. Math. Anal., 57 (2022), 242–251.
A. C. Schaeffer, Inequalities of A. Markoff and S. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc., 47 (1941), 565–579.
I. Schur, Über algebraische gleichungen, die nur wurzeln mit negativen realteilen besitzen, Z. Angew. Math. Mech., 1 (1921), 307–311.
P. Turán, Über die Ableitung von Polynomen, Compositio Math., 7 (1939), 89–95.
S. L. Wali and W. M. Shah, Some applications of Dubinin’s lemma to rational functions with prescribed poles, J. Math. Anal. Appl., 450 (2017), 769–779.