Recent zbMATH articles in MSC 30C10https://www.zbmath.org/atom/cc/30C102022-05-16T20:40:13.078697ZWerkzeugSome \(L^q\) inequalities for polynomialhttps://www.zbmath.org/1483.300022022-05-16T20:40:13.078697Z"Chanam, Barchand"https://www.zbmath.org/authors/?q=ai:chanam.barchand"Reingachan, N."https://www.zbmath.org/authors/?q=ai:reingachan.n"Devi, Khangembam Babina"https://www.zbmath.org/authors/?q=ai:devi.khangembam-babina"Devi, Maisnam Triveni"https://www.zbmath.org/authors/?q=ai:devi.maisnam-triveni"Krishnadas, Kshetrimayum"https://www.zbmath.org/authors/?q=ai:krishnadas.kshetrimayumSummary: Let \(p(z)\) be a polynomial of degree \(n\). Then Bernstein's inequality is
\[
{\max\limits_{|z|=1} |p'(z)| \leq n \max\limits_{|z|=1} |(z)|}.
\]
For \(q>0\), we denote
\[
\|p\|_q = \left\{\frac{1}{2\pi}\int_0^{2\pi} |p(e^{i\theta})|^q d\theta\right\}^{\frac{1}{q}},
\]
and a well-known fact from analysis gives
\[
\lim_{q\to\infty}\left\{\frac{1}{2\pi}\int_0^{2\pi}\big\vert p(e^{i\theta})\big\vert^q d\theta\right\}^{\frac{1}{q}}=\max_{\vert z\vert=1} \vert p(z)\vert.
\]
Above Bernstein's inequality was extended by \textit{A. Zygmund} [Proc. Lond. Math. Soc., II. Ser. 34, 392--400 (1932; Zbl 0005.35301)] into \(L^q\) norm by proving
\[
\|p'\|_q \leq n\|p\|_q, \quad q \geq 1.
\]
Let \(p(z) = a_0 + \sum_{\nu=\mu}^n a_\nu z^\nu,\) \(1 \leq \mu n,\) be a polynomial of degree n having no zero in \(|z| < k, k \geq 1.\) Then for \(0 < r \leq R \leq k\), \textit{A. Aziz} and \textit{B. A. Zargar} [Math. Inequal. Appl. 1, No. 4, 543--550 (1998; Zbl 0914.30002)] proved
\[
{\max\limits_{|z|=R} |p'(z)| \leq \frac{nR^{\mu-1}(R^\mu + k^\mu)^{\frac{n}{\mu}-1}}{(r^\mu + k^\mu)^{\frac{n}{\mu}}} \max\limits_{|z|=r} |p'(z)|}.
\]
In this paper, we obtain the \(L^q\) version of the above inequality for \(q > 0\). Further, we extend a result of \textit{A. Aziz} and \textit{W. M. Shah} [Math. Inequal. Appl. 7, No. 3, 379--391 (2004; Zbl 1061.30001)] into \(L^q\) analogue for \(q > 0\). Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.Bernstein-Walsh type inequalities for derivatives of algebraic polynomialshttps://www.zbmath.org/1483.300102022-05-16T20:40:13.078697Z"Abdullayev, Fahreddin G."https://www.zbmath.org/authors/?q=ai:abdullayev.fahreddin-g"Gün, Cevahir D."https://www.zbmath.org/authors/?q=ai:gun.cevahir-dSummary: In this work, we study Bernstein-Walsh-type estimations for the derivative of an arbitrary algebraic polynomial in regions with piecewise smooth boundary without cusps of the complex plane. Also, estimates are given on the whole complex plane.Inequalities for the derivative of a polynomial with restricted zeroshttps://www.zbmath.org/1483.300112022-05-16T20:40:13.078697Z"Ahanger, Uzma Mubeen"https://www.zbmath.org/authors/?q=ai:ahanger.uzma-mubeen"Shah, W. M."https://www.zbmath.org/authors/?q=ai:shah.wali-mohammadSummary: For a polynomial \(p(z)\) of degree \(n\), it is known that
\[
\begin{aligned}\max_{|z|=1}|p'(z)|\leq \frac{n}{1+k}\max_{|z|=1}|p(z)|,\end{aligned}
\]
if \(p(z)\neq 0\) in \(|z|<k,k \geq 1\) and
\[
\begin{aligned}\max_{|z|=1}|p'(z)|\geq \frac{n}{1+k}\max_{|z|=1}|p(z)|,\end{aligned}
\]
if \(p(z)\neq 0\) for \(|z|>k\), \(k \leq 1\). In this paper, we assume that there is a zero of multiplicity \(s\), \(s <n\) at a point inside \(|z|=1\) and prove some generalizations and improvements of these inequalities.Inequalities for complex rational functionshttps://www.zbmath.org/1483.300122022-05-16T20:40:13.078697Z"Bidkham, M."https://www.zbmath.org/authors/?q=ai:bidkham.mahmood"Khojastehnezhad, E."https://www.zbmath.org/authors/?q=ai:khojastehnezhad.elaheSummary: For a rational function \(r(z) = p(z)/H(z)\) all zeros of which are in \(|z| \leq 1\), it is known that
\[ \left|r'(z)\right|\ge \frac{1}{2}\left|B'(z)\right|\left|r(z)\right| \text{ for }\left|z\right|=1,\]
where \(H(z)={\prod}_{j=1}^n\left(z-{c}_j\right)\), \(\left|{c}_j\right|>1\), \(n\) is a positive integer, \(B(z) = H^\ast (z)/H(z)\), and \({H}^{\ast }(z)={z}^n\overline{H\left(1/\overline{z}\right)}\). We improve the above-mentioned inequality for the rational function \(r(z)\) with all zeros in \(|z| \leq 1\) and a zero of order \(s\) at the origin. Our main results refine and generalize some known rational inequalities.On an inequality of S. Bernsteinhttps://www.zbmath.org/1483.300132022-05-16T20:40:13.078697Z"Chanam, Barchand"https://www.zbmath.org/authors/?q=ai:chanam.barchand"Devi, Khangembam Babina"https://www.zbmath.org/authors/?q=ai:devi.khangembam-babina"Krishnadas, Kshetrimayum"https://www.zbmath.org/authors/?q=ai:krishnadas.kshetrimayum"Devi, Maisnam Triveni"https://www.zbmath.org/authors/?q=ai:devi.maisnam-triveni"Ngamchui, Reingachan"https://www.zbmath.org/authors/?q=ai:ngamchui.reingachan"Singh, Thangjam Birkramjit"https://www.zbmath.org/authors/?q=ai:singh.thangjam-birkramjitSummary: If \(p(z) = \sum_{\nu=0}^n a_\nu z^\nu\) is a polynomial of degree \(n\) having all its zeros on \(|z|=k\), \(k\leq 1,\) then \textit{N. K. Govil} [J. Math. Phys. Sci. 14, 183--187 (1980; Zbl 0444.30007)] proved that
\[
\max\limits_{|z|=1}|p'(z)| \leq \frac{n}{k^n+k^{n-1}}\max\limits_{|z|=1}|p(z)|.
\]
In this paper, by involving certain coefficients of \(p(z)\), we not only improve the above inequality but also improve a result proved by \textit{K. K. Dewan} and \textit{A. Mir} [Southeast Asian Bull. Math. 31, No. 4, 691--695 (2007; Zbl 1150.30001)].Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surfacehttps://www.zbmath.org/1483.300142022-05-16T20:40:13.078697Z"Charlier, Christophe"https://www.zbmath.org/authors/?q=ai:charlier.christopheSummary: We consider a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the Christoffel-Darboux (CD) kernel, which is built in terms of matrix orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of meromorphic functions in a Riemann surface. If this Riemann surface has genus \(0\), then the matrix valued CD kernel is equivalent to a scalar reproducing kernel of polynomials in the plane. Interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. As an application of our result, we show that the correlation kernel of certain doubly periodic lozenge tiling models admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.Homeomorphisms of \(S^1\) and factorizationhttps://www.zbmath.org/1483.300152022-05-16T20:40:13.078697Z"Dalthorp, Mark"https://www.zbmath.org/authors/?q=ai:dalthorp.mark"Pickrell, Doug"https://www.zbmath.org/authors/?q=ai:pickrell.dougSummary: For each \(n>0\) there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations ``conjugated by \(z\to z^n\)''. We show that these families are free of relations, which determines the structure of ``the group of homeomorphisms of finite type''. We next consider factorization for more robust groups of homeomorphisms. We refer to this as root subgroup factorization (because the factors correspond to root subgroups). We are especially interested in how root subgroup factorization is related to triangular factorization (i.e., conformal welding) and correspondences between smoothness properties of the homeomorphisms and decay properties of the root subgroup parameters. This leads to interesting comparisons with Fourier series and the theory of Verblunsky coefficients.\(L^r\) inequalities for the derivative of a polynomialhttps://www.zbmath.org/1483.300162022-05-16T20:40:13.078697Z"Devi, Khangembam Babina"https://www.zbmath.org/authors/?q=ai:devi.khangembam-babina"Krishnadas, Kshetrimayum"https://www.zbmath.org/authors/?q=ai:krishnadas.kshetrimayum"Chanam, Barchand"https://www.zbmath.org/authors/?q=ai:chanam.barchandSummary: Let \(p(z)\) be a polynomial of degree \(n\) having no zero in \(|z|< k\), \(k\leq 1\), then \textit{N. K. Govil} [Proc. Natl. Acad. Sci. India, Sect. A 50, 50--52 (1980; Zbl 0493.30003)] proved
\[
\max\limits_{|z|=1}|p'(z)|\leq \frac{n}{1+k^n}\max\limits_{|z|=1}|p(z)|,
\]
provided \(|p'(z)|\) and \(|q'(z)|\) attain their maxima at the same point on the circle \(|z|=1\), where
\[
q(z)=z^n\overline{p\left(\frac{1}{\overline{z}}\right)}.
\]
In this paper, we not only obtain an integral mean inequality for the above inequality but also extend an improved version of it into \(L^r\) norm.Bounds for the derivative of a certain class of rational functionshttps://www.zbmath.org/1483.300172022-05-16T20:40:13.078697Z"Gupta, Preeti"https://www.zbmath.org/authors/?q=ai:gupta.preeti"Hans, Sunil"https://www.zbmath.org/authors/?q=ai:hans.sunil"Mir, Abdullah"https://www.zbmath.org/authors/?q=ai:mir.abdullahSummary: In this paper, we shall obtain the bounds for the derivative of a rational function in the supremum norm on the unit circle in both the directions by involving the moduli of all its zeros. The obtained results strengthen some recently proved results.On the location of zeros of polynomialshttps://www.zbmath.org/1483.300182022-05-16T20:40:13.078697Z"Kumar, Prasanna"https://www.zbmath.org/authors/?q=ai:kumar.prasanna-v-k"Dhankhar, Ritu"https://www.zbmath.org/authors/?q=ai:dhankhar.rituSummary: In this paper, we discuss the necessary and sufficient conditions for a polynomial \(P(z)\) to have all its zeros inside the open unit disc. These results involve two associated polynomials namely, the derivative of the reciprocal polynomial of \(P(z)\) and the reciprocal of the derivative of \(P(z)\). We also derive some generalizations of the classical Theorem of Laguerre.Extremal problems of Bernstein-type and an operator preserving inequalities between polynomialshttps://www.zbmath.org/1483.300192022-05-16T20:40:13.078697Z"Milovanović, G. V."https://www.zbmath.org/authors/?q=ai:milovanovic.gradimir-v"Mir, A."https://www.zbmath.org/authors/?q=ai:mir.abdullah"Hussain, A."https://www.zbmath.org/authors/?q=ai:hussain.adilSummary: Under consideration are the well-known extremal problems of Bernstein-type which relate the uniform norm between polynomials on the unit disk in the plane. We establish a few new inequalities in both directions for the generalized \({\mathcal{B}}_n \)-operator while accounting for the placement of the zeros of the underlying polynomials. Also, we obtain various estimates for the maximum modulus of a polynomial as well as some inequalities of Erdös-Lax type.A Turán-type inequality for polynomialshttps://www.zbmath.org/1483.300202022-05-16T20:40:13.078697Z"Mir, Abdullah"https://www.zbmath.org/authors/?q=ai:mir.abdullahSummary: In this paper, we consider the class of polynomials \(P(z):=\sum \limits_{j=0}^nc_jz^j\) having all zeros in a closed disk \(|z|\le k,\text{where}~ k\ge 1\) and obtain a result that improves and generalizes the results of Govil, Jain and others by using certain coefficients of \(P(z)\).Some integral inequalities for a polynomial with zeros outside the unit diskhttps://www.zbmath.org/1483.300212022-05-16T20:40:13.078697Z"Mir, Abdullah"https://www.zbmath.org/authors/?q=ai:mir.abdullahSummary: The goal of this paper is to generalize and refine some previous inequalities between the \(L^P\)- norms of the \(s^{\mathrm{th}}\) derivative and of the polynomial itself, in the case when the zeros are outside of the open unit disk.A note on a recent result: on the location of the zeros of polynomials (Lacunary type)https://www.zbmath.org/1483.300222022-05-16T20:40:13.078697Z"Mogbademu, Adesanmi Alao"https://www.zbmath.org/authors/?q=ai:mogbademu.adesanmi-alaoSummary: In this paper, we give some corrections and comments about a result which is contained in a published paper in [\textit{D. Tripathi} et al., Nonlinear Funct. Anal. Appl. 24, No. 3, 555--564 (2019; Zbl 1427.30005)].Weighted Chebyshev polynomials on compact subsets of the complex planehttps://www.zbmath.org/1483.300232022-05-16T20:40:13.078697Z"Novello, Galen"https://www.zbmath.org/authors/?q=ai:novello.galen"Schiefermayr, Klaus"https://www.zbmath.org/authors/?q=ai:schiefermayr.klaus"Zinchenko, Maxim"https://www.zbmath.org/authors/?q=ai:zinchenko.maximSummary: We study weighted Chebyshev polynomials on compact subsets of the complex plane with respect to a bounded weight function. We establish existence and uniqueness of weighted Chebyshev polynomials and derive weighted analogs of Kolmogorov's criterion, the alternation theorem, and a characterization due to Rivlin and Shapiro. We derive invariance of the Widom factors of weighted Chebyshev polynomials under polynomial pre-images and a comparison result for the norms of Chebyshev polynomials corresponding to different weights. Finally, we obtain a lower bound for the Widom factors in terms of the Szegő integral of the weight function and discuss its sharpness.
For the entire collection see [Zbl 1479.47003].Number of zeros of polar derivatives of polynomialshttps://www.zbmath.org/1483.300242022-05-16T20:40:13.078697Z"Ramulu, P."https://www.zbmath.org/authors/?q=ai:ramulu.p"Reddy, G. L."https://www.zbmath.org/authors/?q=ai:reddy.g-lakshma(no abstract)Generalizations and sharpenings of certain Bernstein and Turán types of inequalities for the polar derivative of a polynomialhttps://www.zbmath.org/1483.300252022-05-16T20:40:13.078697Z"Singh, Thangjam Birkramjit"https://www.zbmath.org/authors/?q=ai:singh.thangjam-birkramjit"Chanam, Barchand"https://www.zbmath.org/authors/?q=ai:chanam.barchandSummary: Let \(p(z)\) be a polynomial of degree \(n\). The polar derivative of \(p(z)\) with respect to a complex number \(\alpha\) is defined by
\[
D_\alpha p(z)=np(z)+(\alpha-z)p'(z).
\]
If \(p(z)\) has all its zeros in \(|z|\leq k\), \(k\geq 1\), then for \(|\alpha|\geq k\), \textit{A. Aziz} and \textit{N. A. Rather} [Math. Inequal. Appl. 1, No. 2, 231--238 (1998; Zbl 0911.30002)] proved
\[
\max\limits_{|z|=1}|D_\alpha p(z)|\geq n\left(\frac{|\alpha|-k}{1+k^n}\right)\max\limits_{|z|=1}|p(z)|.
\]
In this paper, we first improve as well as generalize the above inequality. Besides, we are able to prove an improvement of a result due to \textit{N. K. Govil} and \textit{G. N. McTume} [Acta Math. Hung. 104, No. 1--2, 115--126 (2004; Zbl 1060.30004)] and also prove an inequality for a subclass of polynomials having all its zeros in \(|z|\geq k\), \(k\leq 1\).Bernstien type inequalities for polynomials with restricted zeroshttps://www.zbmath.org/1483.300262022-05-16T20:40:13.078697Z"Wali, S. L."https://www.zbmath.org/authors/?q=ai:wali.shah-lubna"Shah, W. M."https://www.zbmath.org/authors/?q=ai:shah.wali-mohammadSummary: In this paper we prove results by using a simple but elegant techniques to improve and strengthen known generalisations and refinements of some widely known polynomial inequalities and thereby deduce useful corollaries from these results.Some biorthogonal polynomials arising in numerical analysis and approximation theoryhttps://www.zbmath.org/1483.420162022-05-16T20:40:13.078697Z"Lubinsky, D. S."https://www.zbmath.org/authors/?q=ai:lubinsky.doron-s"Sidi, A."https://www.zbmath.org/authors/?q=ai:sidi.avramThe aim of the paper under review is the study of some biorthogonal polynomials that arise in certain topics of Numerical Analysis and Approximation Theory such as Numerical Integration and Convergence Acceleration.
Throughout their work the authors discuss the most general form of biorthogonality, that is, involving two families of polynomials orthogonal with respect to certain measures.They provide a survey of these polynomials, biorthogonal with respect different measures such as the logarithm or exponentials.
The authors also discuss the positivity of the weights in the interpolatory quadrature formulas generated by these biorthogonal polynomials. Finally they show the application of the potential theory, powerful tool in so many problems involving polynomials, to the study of some topics of biorthogonal polynomials such as aymptotics and zero distributions.
Reviewer: María-José Cantero (Zaragoza)