Recent zbMATH articles in MSC 30C15 https://www.zbmath.org/atom/cc/30C15 2022-05-16T20:40:13.078697Z Werkzeug Some $$L^q$$ inequalities for polynomial https://www.zbmath.org/1483.30002 2022-05-16T20:40:13.078697Z "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.kshetrimayum Summary: 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. Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applications https://www.zbmath.org/1483.30008 2022-05-16T20:40:13.078697Z "Do, Yen Q." https://www.zbmath.org/authors/?q=ai:do.yen-q Summary: This paper seeks to further explore the distribution of the real roots of random polynomials with non-centered coefficients. We focus on polynomials where the typical values of the coefficients have power growth and count the average number of real zeros. Almost all previous results require coefficients with zero mean, and it is non-trivial to extend these results to the general case. Our approach is based on a novel comparison principle that reduces the general situation to the mean-zero setting. As applications, we obtain new results for the Kac polynomials, hyperbolic random polynomials, their derivatives, and generalizations of these polynomials. The proof features new logarithmic integrability estimates for random polynomials (both local and global) and fairly sharp estimates for the local number of real zeros. Inequalities for the derivative of a polynomial with restricted zeros https://www.zbmath.org/1483.30011 2022-05-16T20:40:13.078697Z "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-mohammad Summary: 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 functions https://www.zbmath.org/1483.30012 2022-05-16T20:40:13.078697Z "Bidkham, M." https://www.zbmath.org/authors/?q=ai:bidkham.mahmood "Khojastehnezhad, E." https://www.zbmath.org/authors/?q=ai:khojastehnezhad.elahe Summary: 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. Bernstein https://www.zbmath.org/1483.30013 2022-05-16T20:40:13.078697Z "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-birkramjit Summary: 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)]. $$L^r$$ inequalities for the derivative of a polynomial https://www.zbmath.org/1483.30016 2022-05-16T20:40:13.078697Z "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.barchand Summary: 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 functions https://www.zbmath.org/1483.30017 2022-05-16T20:40:13.078697Z "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.abdullah Summary: 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 polynomials https://www.zbmath.org/1483.30018 2022-05-16T20:40:13.078697Z "Kumar, Prasanna" https://www.zbmath.org/authors/?q=ai:kumar.prasanna-v-k "Dhankhar, Ritu" https://www.zbmath.org/authors/?q=ai:dhankhar.ritu Summary: 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 polynomials https://www.zbmath.org/1483.30019 2022-05-16T20:40:13.078697Z "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.adil Summary: 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 polynomials https://www.zbmath.org/1483.30020 2022-05-16T20:40:13.078697Z "Mir, Abdullah" https://www.zbmath.org/authors/?q=ai:mir.abdullah Summary: 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 disk https://www.zbmath.org/1483.30021 2022-05-16T20:40:13.078697Z "Mir, Abdullah" https://www.zbmath.org/authors/?q=ai:mir.abdullah Summary: 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/1483.30022 2022-05-16T20:40:13.078697Z "Mogbademu, Adesanmi Alao" https://www.zbmath.org/authors/?q=ai:mogbademu.adesanmi-alao Summary: 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)]. Number of zeros of polar derivatives of polynomials https://www.zbmath.org/1483.30024 2022-05-16T20:40:13.078697Z "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) Bernstien type inequalities for polynomials with restricted zeros https://www.zbmath.org/1483.30026 2022-05-16T20:40:13.078697Z "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-mohammad Summary: 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. The fundamental theorem of algebra and Liouville's theorem geometrically revisited https://www.zbmath.org/1483.30027 2022-05-16T20:40:13.078697Z "Almira, Jose Maria" https://www.zbmath.org/authors/?q=ai:almira.jose-maria "Romero, Alfonso" https://www.zbmath.org/authors/?q=ai:romero.alfonso Summary: If $$f(z)$$ is either a polynomial with no zeroes or a bounded entire function, then a Riemannian metric $$g_f$$ is constructed on the complex plane $$\mathbb{C}$$. This metric $$g_f$$ is shown to be flat and geodesically complete. Therefore, the Riemannian manifold $$(\mathbb{C}, g_f)$$ must be isometric to $$(\mathbb{C}, |dz|^2)$$, which implies that $$f(z)$$ is a constant. Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials https://www.zbmath.org/1483.30028 2022-05-16T20:40:13.078697Z "Beliaev, D." https://www.zbmath.org/authors/?q=ai:beliaev.dmitri-b "Muirhead, S." https://www.zbmath.org/authors/?q=ai:muirhead.stephen "Wigman, I." https://www.zbmath.org/authors/?q=ai:wigman.igor Summary: Beginning with the predictions of Bogomolny-Schmit for the random plane wave, in recent years the deep connections between the level sets of smooth Gaussian random fields and percolation have become apparent. In classical percolation theory a key input into the analysis of global connectivity are scale-independent bounds on crossing probabilities in the critical regime, known as Russo-Seymour-Welsh (RSW) estimates. Similarly, establishing RSW-type estimates for the nodal sets of Gaussian random fields is a major step towards a rigorous understanding of these relations. The Kostlan ensemble is an important model of Gaussian homogeneous random polynomials. The nodal set of this ensemble is a natural model for a `typical' real projective hypersurface, whose understanding can be considered as a statistical version of Hilbert's 16th problem. In this paper we establish RSW-type estimates for the nodal sets of the Kostlan ensemble in dimension two, providing a rigorous relation between random algebraic curves and percolation. The estimates are uniform with respect to the degree of the polynomials, and are valid on all relevant scales; this, in particular, resolves an open question raised recently by Beffara-Gayet. More generally, our arguments yield RSW estimates for a wide class of Gaussian ensembles of smooth random functions on the sphere or the flat torus. Paatero's $$V(k)$$ space and a claim by Pinchuk https://www.zbmath.org/1483.30034 2022-05-16T20:40:13.078697Z "Andreev, Valentin V." https://www.zbmath.org/authors/?q=ai:andreev.valentin-v "Bekker, Miron B." https://www.zbmath.org/authors/?q=ai:bekker.miron-b "Cima, Joseph A." https://www.zbmath.org/authors/?q=ai:cima.joseph-a Summary: In this article we obtain a factorization theorem for the functions in Paatero's $$V(k)$$ space. We bring attention to a significant result of Pinchuk which unfortunately is false. This result relates measures associated to functions in $$V(k)$$ and an integral representation theorem for such functions. We prove necessary and sufficient conditions for a wide class of functions (in particular, the polynomials) to belong to the Paatero class based on the geometry of their critical points, and obtain explicit representation of the measures associated to a wide class of such polynomials that includes the Suffridge polynomials. On the number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficients https://www.zbmath.org/1483.30061 2022-05-16T20:40:13.078697Z "Nguyen, Thu Hien" https://www.zbmath.org/authors/?q=ai:nguyen.thu-hien-thi "Vishnyakova, Anna" https://www.zbmath.org/authors/?q=ai:vishnyakova.anna-m Summary: For an entire function $$f(z)=\sum^\infty_{k=0}a_kz^k$$, $$a_k>0$$, we define the sequence of the second quotients of Taylor coefficients $$Q:=\bigg(\frac{a^2_k}{a_{k-1}a_{k+1}}\bigg)^\infty_{k=1}$$. We find new necessary conditions for a function with a non-decreasing sequence $$Q$$ to belong to the Laguerre-Pólya class of type I. We also estimate the possible number of non-real zeros for a function with a non-decreasing sequence $$Q$$.