On some inequalities concerning polynomials with restricted zeros

• Autorzy: Bhat Aijaz , Bhat Tanveer , Rather N. A.

Identyfikatory

DOI

10.1515/jaa-2023-0160

• Języki publikacji: EN

Abstrakty

EN

In this work we prove some new Bernstein-type inequalities for the polynomials with restricted zeros. Our results strengthen some recently proved Erdős-Lax and Turán-type inequalities by Kumar and others.We further extend the obtained results to the polar derivative of a polynomial.

Słowa kluczowe

EN

polynomials; inequalities; complex domain

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2024
• Tom: Vol. 30, nr 2
• Strony: 371--378
• Opis fizyczny: Bibliogr. 18 poz.

Twórcy

autor

Bhat Aijaz

• Department of Mathematics, Government Degree College Drass, Drass, Kargil Ladakh-194102, India

• https://orcid.org/0009-0005-7027-5098

autor

Bhat Tanveer

• Department of Mathematics, University of Kashmir, Srinagar-190006, India

autor

Rather N. A.

• Department of Mathematics, University of Kashmir, Srinagar-190006, India

Bibliografia

• [1] A. Aziz, Inequalities for the derivative of a polynomial, Proc. Amer. Math. Soc. 89 (1983), no. 2, 259-266.
• [2] A. Aziz, Inequalities for the polar derivative of a polynomial, J. Approx. Theory 55 (1988), no. 2, 183-193.
• [3] A. Aziz and N. Ahmad, Inequalities for the derivative of a polynomial, Proc. Indian Acad. Sci. Math. Sci. 107 (1997), no. 2, 189-196.
• [4] A. Aziz and N. A. Rather, A refinement of a theorem of Paul Turán concerning polynomials, Math. Inequal. Appl. 1 (1998), no. 2, 231-238.
• [5] A. Aziz and N. A. Rather, Inequalities for the polar derivative of a polynomial with restricted zeros, Math. Balkanica (N. S.) 17 (2003), no. 1-2, 15-28.
• [6] A. Aziz and N. A. Rather, Some Zygmund type Lq inequalities for polynomials, J. Math. Anal. Appl. 289 (2004), no. 1, 14-29.
• [7] S. Bernstein, Sur la meilleure approximation de |x| par des polynomes de degrés donnés, Acta Math. 37 (1914), no. 1, 1-57.
• [8] K. K. Dewan, J. Kaur and A. Mir, Inequalities for the derivative of a polynomial, J. Math. Anal. Appl. 269 (2002), no. 2, 489-499.
• [9] N. K. Govil, On the derivative of a polynomial, Proc. Amer. Math. Soc. 41 (1973), 543-546.
• [10] N. K. Govil, On a theorem of S. Bernstein, Proc. Nat. Acad. Sci. India Sect. A 50 (1980), no. 1, 50-52.
• [11] N. K. Govil and P. Kumar, On Bernstein-type inequalities for the polar derivative of a polynomial, in: Progress in Approximation Theory and Applicable Complex Analysis, Springer Optim. Appl. 117, Springer, Cham (2017), 41-74.
• [12] N. K. Govil and P. Kumar, On sharpening of an inequality of Turán, Appl. Anal. Discrete Math. 13 (2019), no. 3, 711-720.
• [13] 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.
• [14] P. Kumar, On the inequalities concerning polynomials, Complex Anal. Oper. Theory 14 (2020), no. 6, Paper No. 65.
• [15] P. D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509-513.
• [16] M. A. Malik, On the derivative of a polynomial, J. Lond. Math. Soc. (2) 1 (1969), 57-60.
• [17] G. V. Milovanović, A. Mir and A. Hussain, Inequalities of Turán-type for algebraic polynomials, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), no. 4, Paper No. 154.
• [18] P. Turan, Über die Ableitung von Polynomen, Compos. Math. 7 (1939), 89-95.