Some integral inequalities for a polynomial with zeros outside the unit disk

Abdullah Mir

doi:10.2478/ausm-2021-0024

Abstract

The goal of this paper is to generalize and refine some previous inequalities between the L^P- norms of the sth derivative and of the polynomial itself, in the case when the zeros are outside of the open unit disk.

References

[1] V. V. Arestov, On integral inequalities for trigonometric polynomials and their derivatives, Izv. Akad. Nauk. SSSR., Ser. Mat., 45 (1981), 3–22.
Search in Google Scholar Back to article
[2] A. Aziz, Inequalities for the polar derivative of a polynomial, J. Approx. Theory, 55 (1988), 183–193.10.1016/0021-9045(88)90085-8
Search in Google Scholar Back to article
[3] A. Aziz and Q. M. Dawood, Inequalities for a polynomial and its derivative, J. Approx. Theory, 54 (1988), 306–313.10.1016/0021-9045(88)90006-8
Search in Google Scholar Back to article
[4] A. Aziz and N. A. Rather, Some Zygmund type Lq inequalities for polynomials, J. Math. Anal. Appl., 289 (2004), 14–29.10.1016/S0022-247X(03)00530-4
Search in Google Scholar Back to article
[5] A. Aziz and N. A. Rather, New L^q inequalities for polynomials, Math. Ineq. Appl., 1 (1998), 177–191.10.7153/mia-01-16
Search in Google Scholar Back to article
[6] N. G. De-Bruijn, Inequalities concerning polynomials in the complex domain, Nederl. Akad. Wetnesch Proc., 50 (1947), 1265–1272.
Search in Google Scholar Back to article
[7] K. K. Dewan and N. K. Govil, An inequality for self-inverse polynomials, J. Math. Anal. Appl., 95 (1983), 490.10.1016/0022-247X(83)90122-1
Search in Google Scholar Back to article
[8] S. Hans and R. Lal, Generalization of some polynomial inequalities not vanishing in a disk, Analysis Math., 40 (2014), 105–115.10.1007/s10476-014-0202-y
Search in Google Scholar Back to article
[9] P. D. Lax, Proof of a conjecture of P. Erdös on the derivative of a Polynomial, Bull. Amer. Math. Soc., 50 (1944), 509–513.10.1090/S0002-9904-1944-08177-9
Search in Google Scholar Back to article
[10] G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in polynomials, Extremal problems, Inequalities, Zeros, World Scientific, Singapore, 1994.10.1142/1284
Search in Google Scholar Back to article
[11] A. Mir and S. A. Baba, Some integral inequalities for the polar derivative of a polynomial, Anal. Theory Appl., 27 (2011), 340–350.10.1007/s10496-011-0340-z
Search in Google Scholar Back to article
[12] Q. I. Rahman and G. Schmeisser, L^p inequalities for polynomials, J. Approx. Theory, 53 (1988) 26–32.10.1016/0021-9045(88)90073-1
Search in Google Scholar Back to article
[13] A. Zireh, Generalization of certain well-known inequalities for the derivative of polynomials, Analysis Math., 41 (2015), 117–132.10.1007/s10476-015-0109-2
Search in Google Scholar Back to article
[14] A. Zygmund, A remark on conjugate series, Proc. London Math. Soc., 34 (1932), 392–400.10.1112/plms/s2-34.1.392
Search in Google Scholar Back to article

Some integral inequalities for a polynomial with zeros outside the unit disk

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