Geodesic
Sharpening of Turán-type inequality for polynomials
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2024), pp. 39-46 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For the polynomial $P(z) = \displaystyle\sum_{j=0}^{n} c_jz^j$ of degree $n$ having all its zeros in $|z|\leq k$, $ k\geq 1$, V. Jain in \textquotedblleft On the derivative of a polynomial\textquotedblright, Bull. Math. Soc. Sci. Math. Roumanie Tome 59, 339–347 (2016) proved that \begin{align*} \max_{|z|=1}|P^\prime(z)|\geq n\bigg(\frac{|c_0| +|c_n|k^{n+1}}{|c_0|(1+ k^{n+1}) +|c_n|(k^{n+1}+ k^{2n})}\bigg)\max_{|z|=1}|P(z)|. \end{align*} In this paper we strengthen the above inequality and other related results for the polynomials of degree $n\geq 2$.
Mots-clés : polynomial, complex domain.
Keywords: inequality 
@article{IVM_2024_4_a4,
     author = {N. A. Rather and A. Bhat and M. Shafi},
     title = {Sharpening of {Tur\'an-type} inequality for polynomials},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {39--46},
     year = {2024},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2024_4_a4/}
}
TY  - JOUR
AU  - N. A. Rather
AU  - A. Bhat
AU  - M. Shafi
TI  - Sharpening of Turán-type inequality for polynomials
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2024
SP  - 39
EP  - 46
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/IVM_2024_4_a4/
LA  - ru
ID  - IVM_2024_4_a4
ER  - 
%0 Journal Article
%A N. A. Rather
%A A. Bhat
%A M. Shafi
%T Sharpening of Turán-type inequality for polynomials
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2024
%P 39-46
%N 4
%U http://geodesic.mathdoc.fr/item/IVM_2024_4_a4/
%G ru
%F IVM_2024_4_a4
N. A. Rather; A. Bhat; M. Shafi. Sharpening of Turán-type inequality for polynomials. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2024), pp. 39-46. http://geodesic.mathdoc.fr/item/IVM_2024_4_a4/

[1] Bernstein S., “Sur l'ordre de la meilleure approximation des functions continues parles polyn$\hat{o}$mes de degr$\grave{e}$ donné”, Mem. Cl. Sci. Acad. Roy Belg., 4 (1912), 1–103 | MR

[2] Turán P., “Über die Ableitung von Polynomen”, Compositio Math., 7 (1940), 89–95 | MR

[3] Govil N.K., “On the derivative of a polynomial”, Proc. Amer. Math. Soc., 41:2 (1973), 543–546 | DOI | MR

[4] Govil N.K., “Some inequalities for derivatives of polynomials”, J. Approx. Theory, 66:1 (1991), 29–35 | DOI | MR

[5] Milovanović G.V., Mitrinović D.S., Rassias Th.M., Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific Publ. Co., Singapore, 1994 | MR

[6] Rahman Q.I., Schmeisser G., Analytic Theory of Polynomials, Oxford Univ. Press Inc., New York, 2002 | MR

[7] Jain V.K., “On the Derivative of a Polynomial”, Bull. Math. Soc. Sci. Math. Roumanie Tome, 59:4 (2016), 339–347 | MR

[8] Mir A., “A Turán-type inequality for polynomials”, Indian J. Pure Appl. Math., 52 (2021), 911–914 | DOI | MR

[9] Osserman R., “A sharp Schwarz inequality on the boundary”, Proc. Amer. Math. Soc., 128:12 (2000), 3513–3517 | DOI | MR

[10] Frappier C., Rahman Q.I., Ruscheweyh St., “New inequalities for polynomials”, Trans. Amer. Math. Soc., 288:1 (1985), 69–99 | DOI | MR