Sharpening of Tur´an-type inequality for polynomials

We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for complex algebraic polynomials of degree n having no zeros in the open unit disk. This is done by establishing the right Bernstein–Szegő–Videnskii type inequality for real trigonometric polynomials of degree at most n on intervals shorter than the period. The paper is closely related to recent work by B. Nagy and V. Totik. In fact, their asymptotically sharp Bernstein-type inequality for complex algebraic polynomials of degree at most n on subarcs of the unit circle is recaptured by using more elementary methods. Our discussion offers a somewhat new way to see V.S. Videnskii’s Bernstein and Markov type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, two classical polynomial inequalities first published in 1960. A new Riesz–Schur type inequality for trigonometric polynomials is also established. Combining this with Videnskii’s Bernstein-type inequality gives Videnskii’s Markov-type inequality immediately.

Let f(z) be a polynomial of degree n.Given that f(z) has a zero on the circle Izl = p (0 < p < <), we estimate 1/(0)1 and 2*rV027ri/VV<*>)1/2 in terms of max|z| _j l/(z)l.We also consider some other related problems.

Let P(z) be a polynomial of degree n and for a complex number , let DP(z) = nP(z) + (alpha − z) P′(z) denote the polar derivative of the polynomial P(z) with respect to alpha . In this paper we establish some Lr inequalities for the polar derivative of a polynomialwith restricted zeros. Our results not only generalizes some known polynomial inequalities, but also a variety of interesting results can be deduced from these by a fairly uniform procedure.

We describe an approach for finding upper bounds on an ODE dynamical system’s maximal Lyapunov exponent (LE) among all trajectories in a specified set. A minimisation problem is formulated whose infimum is equal to the maximal LE, provided that trajectories of interest remain in a compact set. The minimisation is over auxiliary functions that are defined on the state space and subject to a pointwise inequality. In the polynomial case—i.e. when the ODE’s right-hand side is polynomial, the set of interest can be specified by polynomial inequalities or equalities, and auxiliary functions are sought among polynomials—the minimisation can be relaxed into a computationally tractable polynomial optimisation problem subject to sum-of-squares constraints. Enlarging the spaces of polynomials over which auxiliary functions are sought yields optimisation problems of increasing computational cost whose infima converge from above to the maximal LE, at least when the set of interest is compact. For illustration, we carry out such polynomial optimisation computations for two chaotic examples: the Lorenz system and the Hénon–Heiles system. The computed upper bounds converge as polynomial degrees are raised, and in each example we obtain a bound that is sharp to at least five digits. This sharpness is confirmed by finding trajectories whose leading Lyapunov exponents approximately equal the upper bounds.

Applications of the geometric theory of functions to inequalities for algebraic polynomials are considered. The main attention is paid to constructing a univalent conformal mapping for a given polynomial and to applying the Lebedev and Nehari theorems to this mapping. A new sharp inequality of Bernshtein type for polynomials with restrictions on the growth on a segment or on a circle, inequalities with restrictions on the zeros of the polynomial, and other inequalities are obtained. In particular, classical inequalities by Markov, Bernshtein, and Schur are strengthened. Bibiography: 13 titles.

In this paper we study some weighted polynomial inequalities of Markov type in L2-norm. We use the properties of the system of generalized Hermite polynomials . The polynomials H(α)n (x) are orthogonal in ℝ = (−∞,∞) with respect to the weight function . The classical Hermite polynomials Hn(x) present the special case for α = 0.

Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z) = nP(z) + (α − z)P 0 (z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we present an integral inequality for the polar derivative of a polynomial P(z). Our result includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials.

In this work, we discuss generalizations of the classical Bernstein and\nMarkov type inequalities for polynomials and we present some new inequalities\nfor the $k$th Fr\\'echet derivative of homogeneous polynomials on real and\ncomplex $L_{p}(\\mu)$ spaces. We also give applications to homogeneous\npolynomials and symmetric multilinear mappings in $L_{p}(\\mu)$ spaces. Finally,\nBernstein's inequality for homogeneous polynomials on both real and complex\nHilbert spaces has been discussed.\n

Polynomial inequalities on certain algebraic hypersurfaces

Let P(z) be a polynomial of degree n 1 . In this paper we consider a more general problem of investigating the dependence of maximum of ∣∣∣P(Rz) − αP(z) + β {( R + 1 2 )n − |α| } P(z) ∣∣∣, R 1, on the maximum of |P(z)| on |z| = 1 where α , β are arbitrary complex numbers with |α| 1 , |β | 1 and obtain certain sharp compact generalizations of well-known Bernsteintype polynomial inequalities. Mathematics subject classification (2000): 30C10, 30A10.

Let be a polynomial of degree n and let p′(z) be the derivative of p(z). Then it is well known that and .In this paper we obtain inequalities analogous to these inequalities for polynomials satisfying and with all their coefficients lying in a sector. Our results generalize and sharpen some of the known results.

If is a polynomial of degree n such that in , , then Govil [ Proc. Nat. Acad. Sci., Vol. 50, pp. 50-52, 1980. ] proved &#x0D; provided and attain their maxima at the same point on the circle , where .&#x0D; Equality in the above inequality holds for.&#x0D; In this paper, we extend the above inequality and an improved version of this into polar derivative of a polynomial.

Let p(z) be a polynomial of degree n and for α ∈ C, let Dαp(z):= np(z) + (α − z)p′(z) denote the polar derivative of the polynomial p(z) with respect to α. In this paper, we obtain certain integral inequalities concerning the polar derivative of a polynomial, which besides yielding some interesting results, also includes some well-known theorems as special cases. Moreover we refine some Zygmund type inequalities for the polar derivative of a polynomial and present compact generalizations of some prior results.

Let v z  be a polynomial of degree m having no zero zero in z r  ,r 1 , then for 1  T r , Dewan and Bidkham [J. Math. Anal. Appl., vol. 166, pp. 319-324, 1992] proved The result is best possible and extremal polynomial is In this paper, we extend the above inequality for higher derivatives of p z  , which not only improves as well as generalizes tthis result, but also has interesting implications to earlier known inequalities.

Markov–Bernstein Type Inequalities for Multivariate Polynomials on Sets with Cusps