Rate of growth of polynomials non vanishing inside a circle
DOI:
https://doi.org/10.5269/bspm.64811Resumo
For a polynomial $P(z)=\displaystyle\sum_{v=0}^na_vz^v$ of degree $n$ having all zerosin $|z|\geq k, k\geq 1$ Govil et al.[\emph{ILLINOIS J. of Math.}] proved:
$$|P'(z)|\leq n\dfrac{n|a_0|+k^2|a_1|}{(1+k^2)n|a_0|+2k^2|a_1|}|P(z)|.$$
In this paper besides the refinement of above inequality, we also generalize some well known inequalities.
Referências
1. N. C. Ankeny and T. J. Rivlin, On a theorem of S. Bernstein, Pacific J. Math., 5(1955), 849-852.
2. S. N. Bernstein, Sur la limitation des derivees des polynomes, C. R. Acad. Sci. Paris., 190 (1930), 338-340.
3. V. N. Dubinin, Applications of the Schwarz Lemma to inequalities for entire functions with constraints on zeros. J. Math. Sci.,(N.Y) 143(3)(2007), 3069-3076.
4. C. Frappier, Q. I. Rahman and Rt. St. Ruscheweyh, New inequalities for polynomials, Trans. Amer. Math. Soc., 288(1985), 69-99.
5. N. K. Govil, Q. I. Rahman and G. Schmeisser On the derivative of a polynomial, ILLIONIS J. of Math., 23 (1979).
6. P. D. Lax, Proof of a conjecture of P. Erdos on the derivative of a polynomial, Bull. Amer. Math. Soc., 50(1944), 509- 513.
7. G. V. Milovanovi_c, D. S. Mitrinovic and Th. M. Rassias, Topics in polynomials, Extremal problems, Inequalities, Zeros, World Scientific, Singapore, (1994).
8. M. A. Qazi, On the maximum modulus of Polynomials, Proceedings of the American Math. society 115, (1992).
9. Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, Oxford University Press, New york, 2002.
10. T. Sheil-Small, Complex polynomials, Cambridge University Press, 2002.
11. P. Turan, Uber die ableitung von polynomen, Compos. Math., 7 (1939), 89-95.
12. C. Visser, A simple proof of certain inequalities concerning polynomials, Nederl. Akad. Wetensch. Proc. 48, 276-281, Indag. Math.(7)(1995), 81-86.
2. S. N. Bernstein, Sur la limitation des derivees des polynomes, C. R. Acad. Sci. Paris., 190 (1930), 338-340.
3. V. N. Dubinin, Applications of the Schwarz Lemma to inequalities for entire functions with constraints on zeros. J. Math. Sci.,(N.Y) 143(3)(2007), 3069-3076.
4. C. Frappier, Q. I. Rahman and Rt. St. Ruscheweyh, New inequalities for polynomials, Trans. Amer. Math. Soc., 288(1985), 69-99.
5. N. K. Govil, Q. I. Rahman and G. Schmeisser On the derivative of a polynomial, ILLIONIS J. of Math., 23 (1979).
6. P. D. Lax, Proof of a conjecture of P. Erdos on the derivative of a polynomial, Bull. Amer. Math. Soc., 50(1944), 509- 513.
7. G. V. Milovanovi_c, D. S. Mitrinovic and Th. M. Rassias, Topics in polynomials, Extremal problems, Inequalities, Zeros, World Scientific, Singapore, (1994).
8. M. A. Qazi, On the maximum modulus of Polynomials, Proceedings of the American Math. society 115, (1992).
9. Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, Oxford University Press, New york, 2002.
10. T. Sheil-Small, Complex polynomials, Cambridge University Press, 2002.
11. P. Turan, Uber die ableitung von polynomen, Compos. Math., 7 (1939), 89-95.
12. C. Visser, A simple proof of certain inequalities concerning polynomials, Nederl. Akad. Wetensch. Proc. 48, 276-281, Indag. Math.(7)(1995), 81-86.
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2024-05-08
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