M.A. Komarov. On Borwein’s identity and weighted Turán type inequalities on a closed interval ... P. 127-138

Let $\Pi_n^*$ be the class of algebraic polynomials $P$ of degree $n$ having all zeros on the interval $[-1,1]$ and vanishing at the points $1$ and $-1$. In addition, let $w(x)=1-x^2$. The main result of the paper can be formulated as follows: there is an absolute constant $A>0$ such that
\[ \|P'w^{1-s}\|_{C[-1,1]}>A\sqrt{n}\cdot \sqrt{1-\Delta_P^2}\,\|Pw^{-s}\|_{C[-1,1]} \]
for any $P\in \Pi_n^*$ and $s\in [0,1]$, where $\Delta_P=\inf\big\{d\ge 0\colon \|Pw^{-s}\|_{C[-d,d]}=\|Pw^{-s}\|_{C[-1,1]}\big\}$. This inequality may be interpreted as a weighted analog of P. Turán's classical inequality for the derivative of polynomials with zeros on a closed interval. The proof uses a generalization of an interesting formula of P. Borwein concerning the logarithmic derivative of such polynomials. Our estimate is sharp in the order of the quantity $n$ and complements well-known results of  V.F. Babenko, S.A. Pichugov, S.P. Zhou, and others.

Keywords: logarithmic derivative of a polynomial, weighted Turán inequality

Received September 2, 2021

Revised November 8, 2021

Accepted November 15, 2021

Mikhail Anatol’evich Komarov, Cand. Sci. (Phys.-Math.), Vladimir State University, Vladimir, 600000 Russia, e-mail: kami9@yandex.ru

REFERENCES

1.   Borwein P. The size  of $\{x: r_n'/r_n\ge 1\}$ and lower bounds for $\|e^{-x}-r_n\|$.  J. Approx. Theory, 1982, vol. 36, no. 1, pp. 73–80. doi: 10.1016/0021-9045(82)90072-7 

2.   Macintyre A.J., Fuchs W.H.J. Inequalities for the logarithmic derivatives of a polynomial. J. London Math. Soc., 1940, vol. 15, no. 2, pp. 162–168. doi: 10.1112/jlms/s1-15.3.162 

3.   Komarov M.A. Distribution of the logarithmic derivative of a rational function on the line. Acta Math. Hungar., 2021, vol. 163, no. 2, pp. 623–639. doi: 10.1007/s10474-020-01102-w 

4.   Govorov N.V., Lapenko Yu.P. Lower bounds for the modulus of the logarithmic derivative of a polynomial. Math. Notes, 1978, vol. 23, no. 4, pp. 288–292. doi: 10.1007/BF01786958 

5.   Komarov M.A. Reverse Markov inequality on the unit interval for polynomials whose zeros lie in the upper unit half-disk. Anal. Math., 2019, vol. 45, no. 4, pp. 817–821. doi: 10.1007/s10476-019-0009-y 

6.   Komarov M.A. The Turán-type inequality in the space L0 on the unit interval. Anal. Math., 2021, vol. 47, no. 4, pp. 843–852. doi: 10.1007/s10476-021-0097-3 

7.   Turán  P.  Über die Ableitung von Polynomen. Compos. Math., 1940, vol. 7, no. 89, pp. 89–95. Available on: https://eudml.org/doc/88754 

8.   Varma A.K. An analogue of some inequalities of P. Turán concerning algebraic polynomials having all zeros inside [-1,+1]. Proc. Amer. Math. Soc., 1976, vol. 55, no. 2, pp. 305–309. doi: 10.1090/S0002-9939-1976-0396878-7 

9.   Zhou S.P. An extension of the Turán inequality in $L_p$-space for 0 < p < 1. J. Math. Res. Expos., 1986, vol. 6, no. 2, pp. 27–30. doi: 10.3770/j.issn:1000-341X.1986.02.010 

10.   Glazyrina P.Yu. The Markov brothers inequality in the space $L_0$ on a closed interval. Math. Notes, 2005, vol. 78, no. 1, pp. 53–58. doi: 10.1007/s11006-005-0098-8 

11.   Erdélyi T. Turán-type reverse Markov inequalities for polynomials with restricted zeros. Constr. Approx., 2021, vol. 54, no. 1, pp. 35–48. doi: 10.1007/s00365-020-09509-y 

12.   Babenko V.F., Pichugov S.A. An exact inequality for the derivative of a trigonometric polynomial having only real zeros. Math. Notes, 1986, vol. 39, no. 3, pp. 179–182. doi: 10.1007/BF01170244 

13.   Xiao W., Zhou S. On weighted Turán type inequality. Glas. Mat., III. Ser., 1999, vol. 34, no. 2, pp. 197–202.

14.   Yu D., Wei B. On Turán  type inequality with doubling weights and $A^*$ weights. J. Zheijang Univ. Sci. A, 2005, vol. 6, no. 7, pp. 764–768. doi: 10.1631/jzus.2005.A0764 

15.   Underhill B., Varma A.K. An extension of some inequalities of P. Erdős and P. Turán concerning algebraic polynomials. Acta Math. Hung., 1996, vol. 73, no. 1-2, pp. 1–28. doi: 10.1007/BF00058939 

16.   Wang J.L., Zhou S.P. The weighted Turán type inequality for generalized Jacobi weights. Bull. Aust. Math. Soc., 2002, vol. 66, no. 2, pp. 259–265. doi: 10.1017/S0004972700040107 

17.   Glazyrina P.Yu., Révész Sz.Gy. Turán-Erőd type converse Markov inequalities on general convex domains of the plane in the boundary $L^q$ norm. Proc. Steklov Inst. Math., 2018, vol. 303, pp. 78–104. doi: 10.1134/S0081543818080084 

18.   Baran M. Markov inequality on sets with polynomial parametrization. Ann. Polon. Math., 1994, vol. 60, no. 1, pp. 69–79. doi: 10.4064/ap-60-1-69-79 

Cite this article as: M.A. Komarov. On Borwein’s identity and weighted Turán type inequalities on a closed interval, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 1, pp. 127–138.