Geodesic
Refinement of Erd\"os-Lax inequality for $\mathrm{N}$-operator
Problemy analiza, Tome 14 (2025) no. 1, pp. 42-60.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\mathcal{P}_n$ be the space of all polynomials of degree less than or equal to $n$. In this paper, we establish a refinement of Erdös-Lax inequality in which the classical derivative (as an operator on $\mathcal{P}_n$) is replaced by a $B_n$ operator. The result obtained includes some interesting inequalities as special cases.
Keywords: inequalities, $\mathrm{N}$-operator, polynomials, zeros. 
@article{PA_2025_14_1_a2,
     author = {F. A. Bhat},
     title = {Refinement of {Erd\"os-Lax} inequality for $\mathrm{N}$-operator},
     journal = {Problemy analiza},
     pages = {42--60},
     publisher = {mathdoc},
     volume = {14},
     number = {1},
     year = {2025},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PA_2025_14_1_a2/}
}
TY  - JOUR
AU  - F. A. Bhat
TI  - Refinement of Erd\"os-Lax inequality for $\mathrm{N}$-operator
JO  - Problemy analiza
PY  - 2025
SP  - 42
EP  - 60
VL  - 14
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PA_2025_14_1_a2/
LA  - en
ID  - PA_2025_14_1_a2
ER  - 
%0 Journal Article
%A F. A. Bhat
%T Refinement of Erd\"os-Lax inequality for $\mathrm{N}$-operator
%J Problemy analiza
%D 2025
%P 42-60
%V 14
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PA_2025_14_1_a2/
%G en
%F PA_2025_14_1_a2
F. A. Bhat. Refinement of Erd\"os-Lax inequality for $\mathrm{N}$-operator. Problemy analiza, Tome 14 (2025) no. 1, pp. 42-60. http://geodesic.mathdoc.fr/item/PA_2025_14_1_a2/

[1] Ankeny N. C., Rivilin T. J., “On a theorem of S. Bernstein”, Pacific J. Math., 5:6 (1955), 849–852 | DOI | MR | Zbl

[2] Arestov V. V., “On integral inequalities for trigonometric polynomials and their derivatives”, Math USSR-Izv., 18 (1982), 1–17 | DOI | MR | Zbl

[3] Aziz A., Shah W. M., “Lp inequalities for polynomials with restricted zeros”, Glasnik. Matematićki, 32 (1997), 247–258 https://web.math.pmf.unizg.hr/glasnik/vol_32/no2_08.html | MR | Zbl

[4] Bernstein S., “Sur l'ordre de la meilleure approximation des fonctions continues par des polynômes de degré donné”, Mem. Acad. R. Belg., 4 (1912), 1–103 | MR

[5] Bernstein S. N., “Sur le theoreme de W. Markov”, Trans. Lenin. Ind. Inst., 1938, 8–13

[6] Boas Jr. R. P., Rahman Q. I., “$L^p$ inequalities for polynomials and entire functions”, Arch. Ration. Mech. Anal., 11 (1962), 34–39 | DOI | MR | Zbl

[7] Bruijin N. G., “Inequalities concerning polynomials in the complex domain”, Nederal. Akad. Wetensch. Proc., 50 (1947), 1265–1272 | MR

[8] De Bruijn N. G., Springer T. A., “On the zeros of composite polynomials”, Proceedings of the Section of Sciences of the Koninklijke Nederlande Akademie van Wetenschappen te Amsterdam, 50:7-8 (1947), 895–903 | Zbl

[9] Govil N. K., Liman A., Shah W. M., “Some inequalities concerning derivative and maximum modulus of polynomials”, Austr. J. Math. Anal. Appl., 8 (2011), 1–8 https://ajmaa.org/cgi-bin/paper.pl?string=v8n1/V8I1P7.tex | Zbl

[10] Hardy G. H., “The mean value of the modulus of an analytic function”, Proc. Lond. math. Soc., 14 (1915), 269–277 | DOI | MR

[11] Kompaneets E. G., Zybina L. G., “Smirnov and Bernstein-type inequalities, taking into account higher order coefficients and free terms of polynomials”, Probl. Anal. Issues Anal., 13(31):1 (2024), 3–23 | DOI | MR | Zbl

[12] Lax P. D., “Proof of a conjecture of P. Erdös on the derivative of a polynomial”, Bull. Amer. Math. Soc., 50 (1944), 509–513 | DOI | MR | Zbl

[13] Markov A. A., “On a problem posed by D. I. Mendeleev”, Izv. Akad. Nauk. St. Petersburg, 62 (1889), 1–24 (in Russian)

[14] Mendeleev D. I., Investigation of aqueous solutions by specific gravity, Tip. V. Demakova, St. Petersburg, 1887 (in Russain)

[15] Milovanovic G. V., Mitrinovic D. S., Rassias Th. M., Topics in Polynomials: Extremal Properties, Inequalities, Zero, World Scientific Publishing co., Singapore, 1994 | MR

[16] Mir A., “Comparison inequalities of Bernstein-type between polynomials with restricted zeros”, Appl. Anal. Discrete Math., 16:1 (2022), 55–65 | DOI | MR | Zbl

[17] Mir A., Rather N. A., Dar I., “Zygmund type integral inequalities for complex polynomials”, Mediterr. J. Math., 20 (2023), 25 | DOI | MR | Zbl

[18] Rahman Q. I., “Functions of exponential type”, Trans. Amer. Math. Soc., 135 (1969), 295–309 | DOI | MR | Zbl

[19] Rassias Th. M., “On certain properties of polynomials and their derivative”, Topics in Mathematical Analysis, World Scientific Publishing Company, Singapore, 1989, 758–802 | DOI | MR

[20] Rahman Q. I., Schmeisser G., Les inegalites de Markoff et de Bernstein, Presses Univ. Montreal, Montreal, Quebec, 1983 | MR | Zbl

[21] Rahman Q. I., Schmeisser G., “$L^p$ inequalities for polynomials”, J. Approx. Theory, 53 (1988), 26–32 | DOI | MR | Zbl

[22] Rahman Q. I., Schmeisser G., Analytic Theory of Polynomials, Oxford University press, 2000 | MR

[23] Rather N. A., Dar I., Gulzar S., “On the zeros of certain composite polynomials and an operator preserving inequalities”, Ramanujan J., 54 (2021), 605–612 | DOI | MR | Zbl

[24] Rather N. A., Shah M. A., “On an operator preserving $L_p$ inequalities between polynomials”, J. Math. Anal. Appl., 399 (2013), 422–432 | DOI | MR | Zbl

[25] Rather N. A., Wani N., Bhat A., “Integral mean estimate with restricted zeros for polynomials”, Probl. Anal. Issues Anal., 13(31):3 (2024), 101–117 | DOI | MR

[26] Schaeffer A. C., “Inequalities of A. Markoff and S. N. Bernstein for polynomials and related functions”, Bull. Amer. Math. Soc., 47 (1941), 565–579 | DOI | MR | Zbl

[27] Wali S. L., “An integral estimate for an operator preserving inequalities between polynomials”, Probl. Anal. Issues Anal., 9 (27):3 (2020), 137–147 | DOI | MR | Zbl

[28] Zygmund A., “A remark on conjugate series”, Proc. Lond. Math. Soc., s2-34:1 (1932), 392–400 | DOI | MR | Zbl