On Bernstein's inequality for entire functions of exponential type

It was shown by S.N. Bernstein that if f is an entire function of exponential type τ such that |f(x)|?M for −∞<x<∞, then |f^′(x)|?Mτ for −∞<x<∞. If p is a polynomial of degree at most n with |p(z)|?M for |z|=1, then f(z):=p(eiz) is an entire function of exponential type n with |f(x)|?M on the real axis. Hence, by the just mentioned inequality for functions of exponential type, |p^′(z)|?Mn for |z|=1. Lately, many papers have been written on polynomials p that satisfy the condition znp(1/z)≡p(z). They do form an intriguing class. If a polynomial p satisfies this condition, then f(z):=p(eiz) is an entire function of exponential type n that satisfies the condition f(z)≡einzf(−z). This led Govil [N.K. Govil, Lp inequalities for entire functions of exponential type, Math. Inequal. Appl. 6 (2003) 445-452] to consider entire functions f of exponential type satisfying f(z)≡eiτzf(−z) and find estimates for their derivatives. In the present paper we present some additional observations about such functions.     

On the coefficients of a bounded univalent analytic function

LetF _M denote the class of univalent analytic functionsf in |z|<1 with the expansionf (z)=z+a ₂ z ²+a ₃ z ³+... and |f(z)|?M in |z|<1. In this note I derive a rough bound for alln-th coefficients and a more accurate bound for all the third coefficients of functionsf belonging toF _M.     

Solutions of functional equations having bounded variation

Summary. We prove that a solution f of the functional equation¶¶f(t)=h(t,y,f(g₁(t,y)),...,f(g_n(t,y))) f(t)=h(t,y,f(g_1(t,y)),\dots,f(g_n(t,y))) ¶ having locally bounded variation is a C^￥ {\cal C}^\infty -function.     

ON Sums of Products of Polynomials

Abstract

Given a polynomial P(t₁ ,…, t _n) = σ a_a t_a ^a1 …t_n ^an in several variables, we consider the p-norms |P|_p = (σ |a_a | ^p )^1/p (1≥ p < ∞) and |p|_∞ = max |a_a |. Our goal is to establish a generalization to the p-norms (1 ≥ p ≥ ∞) of a theorem originally obtained by P. Enflo for the l-norm.     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Majorization of polynomials on the plane

We prove that for every χ[−1, 1] and every real algebraic polynomial f of degree n such that |f(t): 1 on [−1, 1], the following inequality takes place on the complex plane |f(x+iy)||T_n(1+iy)|,−∞y∞ where T_n is the Tchebycheff polynomial. This implies easily Vladimir Markov inequality.     

The mean curvature of the influence surface of wave equation with sources on a moving surface

The mean curvature of the influence surface of the space–time point ( x , t) appears in linear supersonic propeller noise theory and in the Kirchhoff formula for a supersonic surface. Both these problems are governed by the linear wave equation with sources on a moving surface. The influence surface is also called the Σ‐surface in the aeroacoustic literature. This surface is the locus, in a frame fixed to the quiescent medium, of all the points of a radiating surface f( x , t)=0 whose acoustic signals arrive simultaneously to an observer at position x and at the time t. Mathematically, the Σ‐surface is produced by the intersection of the characteristic conoid of the space–time point ( x , t) and the moving surface. In this paper, we derive the expression for the local mean curvature of the Σ‐surface of the space–time point ( x , t) for a moving rigid or deformable surface f( x , t)=0. This expression is a complicated function of the geometric and kinematic parameters of the surface f( x , t)=0. Using the results of this paper, the solution of the governing wave equation of high‐speed propeller noise radiation as well as the Kirchhoff formula for a supersonic surface can be written as very compact analytic expressions. Copyright © 1999 John Wiley & Sons, Ltd.     

On the total curvatures of a tame function

Given a definable function f: ℝ ⁿ ↦ ℝ, enough differentiable, we study the continuity of the total curvature function t → K(t), total curvature of the level f ⁻¹(t), and the total absolute curvature function t → |K|(t), total absolute curvature of the level f ⁻¹(t). We show they admits at most finitely many discontinuities. Partially supported by the European research network IHP-RAAG contract number HPRN-CT-2001-00271 and partially supported by Deutsche Forschungs-Gemeinschaft in the Priority Program Global Differential Geometry.     

The symmetrized Sine addition formula

As a continuation of An and Yang (Integral Equ Oper Theory 66:183–195, 2010) in this paper, the symmetrized Sine addition formula

On trigonometric polynomials least deviating from zero

We give a solution of the problem about trigonometric polynomials with a given leading harmonic and least deviating from zero in measure; more precisely, with respect to the functional μ(f _n ) = mes {t ∈ [0, 2π]: |f _n (t)| ≥ 1}. We give a solution of a related problem about the minimal value over compact sets (from the real line) of a given measure of least uniform deviation from zero on a compact set for trigonometric polynomials with a fixed leading harmonic. Published in Russian in Doklady Akademii Nauk, 2009, Vol. 425, No. 6, pp. 733–736. Presented by Academician A.M. Il’in November 13, 2008 The article was translated by the authors.     

On the Semicontinuity of Curvature Integrals

Let H_j(K, ·) be the j – th elementary symmetric function of the principal curvatures of a convex body K in Euclidean d – space. We show that the functionals ∫_bd f(H_j(K, x)) dℋ︁^d—1(x) depend upper semicontinuously on K, if f : [0, ∞) is concave, lim_t→0f(t) = 0, and lim_t→∞f(t)/t = 0. An analogous statement holds for integrals of elementary symmetric functions of the principal radii of curvature.     

An algorithm of MCMC method for solving F(X) =0

An algorithm of continuous stage-space MCMC method for solving algebra equation f(x)=0 is given. It is available for the case that the sign of f(x) changes frequently or the derivative f′(x) does not exist in the neighborhood of the root, while the Newton method is hard to work. Let n be the number of random variables created by computer in our algorithm. Then after m=O(n) transactions from the initial value x ₀,x* can be got such that |f(x*)|<e ^−cm |f(x ₀)| by choosing suitable positive constant c. An illustration is also given with the discussion of convergence by adjusting the parameters in the algorithm. Supported by the National Natural Science Foundation of China (70171008).     

Existence results for non-autonomous multiple-fragmentation models

We investigate an initial-value problem modelling fragmentation processes where particles split into two or more pieces at a rate, γ, that not only depends on the sizes of the particles involved but also on time. The existence of non-negative, mass-conserving solutions is established by considering a truncated version of an associated non-autonomous abstract Cauchy problem. The latter has solutions of the form u(t)=U_n(t,t₀)f, t⩾t₀, where f is the known data at some fixed time t₀⩾0 and {U_n(t,s)} is a uniformly continuous evolution system. A limit evolution system {U(t,s)} is shown to exist. Depending on the form of the known data f at time t₀, the scalar-valued function u, obtained from the limit evolution system via u(x, t)=[U(t, t₀)f](x) for a.e. x>0, t⩾t₀, is a solution of either the original initial-value problem or an integral version of this problem. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

A remark on the genericity of multiplicity results for forced oscillations on manifolds

We discuss the genericity of some multiplicity results for periodically perturbed autonomous first- and second-order ODEs on manifolds.?In particular, the genericity of the following property is investigated: if the differentiable manifold M is compact, then the equation _π=h(x,)+f(t,x,) on M has |χ(M)| geometrically distinct T-periodic solutions for any small enough T-periodic perturbing function f. Received: January 24, 2000; in final form: January 16, 2001?Published online: March 19, 2002     

On Sommerfeld representation and uniqueness in scattering by wedges

We consider a non‐stationary scattering of plane waves by a wedge. We prove the Sommerfeld‐type representation and uniqueness of solution to the Cauchy problem in appropriate functional spaces developing the general method of complex characteristics (Math. USSR Sb. 1973; 21 (1):91–135, Moscow Univ. Math. Bull. 1974; 29 (2):140–145, Oper. Theory Adv. Appl. 1992; 57 :171–183, Am. Math. Soc. Transl. (2) 2002; 206 :125–159). Copyright © 2004 John Wiley & Sons, Ltd.     

Some New Applications of the Subspace Theorem

We present some applications of the Subspace Theorem to the investigation of the arithmetic of the values of Laurent series f(z) at S-unit points. For instance we prove that if f(q ⁿ ) is an algebraic integer for infinitely many n, then h(f(q ⁿ )) must grow faster than n. By similar principles, we also prove diophantine results about power sums and transcendency results for lacunary series; these include as very special cases classical theorems of Mahler. Our arguments often appear to be independent of previous techniques in the context.     

Toeplitz Operators with Uniformly Continuous Symbols

Let T _f be a Toeplitz operator on the Segal–Bargmann space or the standard weighted Bergman space over a bounded symmetric domain \({\Omega \subset {\bf C}^n}\) with possibly unbounded symbol f. Combining recent results in Bauer et al. (J. Funct. Anal. 259:57–78, 2010), Bauer et al. (J. reine angew. Math. doi: 10.1515/crelle-2015-0016), Issa (Integr. Equ. Oper. Theory 70:569–582, 2011) we show that in the case of uniformly continuous symbols f with respect to the Euclidean metric on C ⁿ and the Bergman metric on \({\Omega}\), respectively, the operator T _f is bounded if and only if f is bounded. Moreover, T _f is compact if and only if f vanishes at the boundary of \({\Omega.}\) This observation substantially extends a result in Coburn (Indiana Univ. Math. J. 23:433–439, 1973).     

Radial solutions to the wave equation

We study some boundedness properties of radial solutions to the Cauchy problem associated to the wave equation (∂ _t ²-▵ _x )u(t,x)=0 and meanwhile we give a new proof of the solution formula. Received: July 7, 1998?Published online: March 19, 2002     

A simple GAP-canceling algorithm for the generalized maximum flow problem

We give a simple primal algorithm for the generalized maximum flow problem that repeatedly finds and cancels generalized augmenting paths (GAPs). We use ideas of Wallacher (A generalization of the minimum-mean cycle selection rule in cycle canceling algorithms, 1991) to find GAPs that have a good trade-off between the gain of the GAP and the residual capacity of its arcs; our algorithm may be viewed as a special case of Wayne’s algorithm for the generalized minimum-cost circulation problem (Wayne in Math Oper Res 27:445–459, 2002). Most previous algorithms for the generalized maximum flow problem are dual-based; the few previous primal algorithms (including Wayne in Math Oper Res 27:445–459, 2002) require subroutines to test the feasibility of linear programs with two variables per inequality (TVPIs). We give an O(mn) time algorithm for finding negative-cost GAPs which can be used in place of the TVPI tester. This yields an algorithm with O(m log(mB/ε)) iterations of O(mn) time to compute an ε-optimal flow, or O(m ² log (mB)) iterations to compute an optimal flow, for an overall running time of O(m ³ nlog(mB)). The fastest known running time for this problem is , and is due to Radzik (Theor Comput Sci 312:75–97, 2004), building on earlier work of Goldfarb et al. (Math Oper Res 22:793–802, 1997). David P. Williamson is supported in part by an IBM Faculty Partnership Award and NSF grant CCF-0514628.     

Wave equations with time-dependent spatial operators of higher order

We study the initial-boundary value problem for ?_t²u(t,x)+A(t)u(t,x)+B(t)?_tu(t,x)=f(t,x) on [0,T]×Ω(Ω??ⁿ) with a homogeneous Dirichlet boundary condition; here A(t) denotes a family of uniformly strongly elliptic operators of order 2m, B(t) denotes a family of spatial differential operators of order less than or equal to m, and u is a scalar function. We prove the existence of a unique strong solution u. Furthermore, an energy estimate for u is given.