The Bicanonical Map of Surfaces with pg = 0 and K2 >= 7

A minimal surface of general type with p_g(S) = 0 satisfies 1 K² 9, and it is known that the image of the bicanonical map is a surface for , whilst for , the bicanonical map is always a morphism. In this paper it is shown that is birationalif , and that the degree of is at most 2 if or By presenting two examples of surfaces S with and 8 and bicanonical map of degree 2, it is alsoshown that this result is sharp. The example with is, to our knowledge, a new example of a surfaceof general type with p_g = 0. The degree of is also calculated for two other known surfacesof general type with p_g = 0 and . In both cases, the bicanonical map turns out to be birational.     

Oscillation Criteria for First-Order Delay Equations

This paper is concerned with the oscillatory behaviour of first-orderdelay differential equations of the form (1) where is non-decreasing, (t)< t for t t₀ and . Let the numbers k andL be defined by It is proved here that when L < 1 and 0 < k 1/e all solutionsof equation (1) oscillate in several cases in which the condition holds, where ₁ is the smaller root of the equation = e^k. 2000Mathematics Subject Classification 34K11 (primary); 34C10 (secondary).     

A Radial Uniqueness Theorem for Sobolev Functions

We show that continuous functions u in the Sobolev space , 1 < p n, which have the limitzero in a certain weak sense in a set of positive p-capacityon B with where B is the open unit ball of Rⁿ and for 0 > > , are identically zero. Conversely, we produce for each 1 > p n and each positive a non-constant function u in , continuous in , and a compact set EB of positive p-capacity such that u = 0 in E and the aboveinequality holds with exponent p – l + .     

A Schwarz Lemma for the Symmetrized Bidisc

Let be an analytic function from D to the symmetrized bidisc We show that if (0) = (0,0) and () = (s, p) in the interiorof , then Moreover, the inequality is sharp: we give an explicit formulafor a suitable in the event that the inequality holds withequality. We show further that the inverse hyperbolic tangentof the left-hand side of the inequality is equal to both theCaratheodory distance and the Kobayashi distance from (0,0)to (s, p) in int      

Paley's Inequality for the Jacobi Expansions

Let be an analytic function in the unit disc satisfying Then, which is familiar as Paley's inequality. In this paper, an analogue of this inequality withrespect to the Jacobi expansions is established.     

Uniqueness of a Nonharmonic Trigonometric Series under an Exponential Growth Condition

We prove uniqueness for the nonharmonic trigonometric series under the weaker condition (*) where , for some 0 < < 1. In other words, if satisfies the above condition (*), and if , then a_k = 0 for all k = 0, 1,.... Finally, we statean improvement of Zygmund's uniqueness result as a corollary.     

Bernstein-Type Inequalities for Linear Combinations of Shifted Gaussians

Let P_n be the collection of all polynomials of degree at mostn with real coefficients. A subtle Bernstein-type extremal problemis solved by establishing the inequality for all , and m= 1, 2, ..., where c is an absolute constant and Some related inequalities and direct and inversetheorems about the approximation by elements of in L_q (R) are also discussed. 2000 Mathematics SubjectClassification 41A17 (primary).     

On the Best Possible Constants in the Khintchine Inequality for P >= 3

In the paper the best constants in the Khintchine inequalityfor p 3 are found, namely where . This gives stronger estimates than the hitherto Known ones forthe constants depend on n.     

Estimates of Eigenfunctions for One Class of Boundary Conditions

Requirements for boundary conditions are found which imply sharpby order estimates in L-metrics of eigenfunctions of the ordinarydifferential operator It is shown that the inequalities for the eigenfunctions {y} yield certain restrictions on thespectrum. 1991 Mathematics Subject Classification 34L05.     

On Approximately Midconvex Functions

A real-valued function f defined on an open, convex set D ofa real normed space is called (, )-midconvex if it satisfies The main result of the paper states that if f is locally boundedfrom above at a point of D and is (, )-midconvex, then it satisfiesthe convexity-type inequality where : [0, 1] R is a continuous function satisfying The particular case = 0 of this result is due to Ng and Nikodem(Proc. Amer. Math. Soc. 118 (1993) 103–108), while thespecialization = = 0 yields the theorem of Bernstein and Doetsch(Math. Ann. 76 (1915) 514–526). 2000 Mathematics SubjectClassification 26A51, 26B25.     

On the Best Constant in Weighted Inequalities for Riemann-Liouville Integrals

For 1 k < and 1 p q , the problem of finding the bestconstant C_pq in the weighted inequality involving the Riemann-Liouville integrals of theform is considered.     

A Combinatorial Application of the Maximal Ergodic Theorem

Let X be a compact space,µ a Borel probability measureon X, T: X X a measure preserving continuous transformationand g: X R a continuous function. Then for some yX, This Lemma is used to give an alternative proof of a resultby Ruzsa [6], which implies the following extension of a resultof Bergelson [1]. If E N satisfies then there exists a set N such that n–1|[1,n]| (E) for all, n 1, and any finite subset{₁, ... _k} satisfies Ø. 7 Moria St., Ramat Hasharon, Israel     

Some Relations Between Packing Premeasure and Packing Measure

Let K be a compact subset of Rⁿ, 0 s n. Let , P^s denote s-dimensional packing premeasure andmeasure, respectively. We discuss in this paper the relationbetween and P^s. We prove:if , then ; and if , then for any > 0, there exists a compact subset F of K such that and P^s(F) P^s(K) – .1991 Mathematics Subject Classification 28A80, 28A78.     

Generalized Logarithmic Derivative Estimates of Gol'dberg-Grinshtein Type

For f meromorphic in the complex plane and meromorphic in theunit disc, sharp upper bounds are obtained for and where k andj are integers satisfying k > j 0. The results generalizethe logarithmic derivative estimate due to Gol'dberg and Grinshteinto derivatives of higher order. 2000 Mathematics Subject Classification30D35.     

On the Discrete Hardy Inequality

Necessary and sufficient conditions for the boundedness of thediscrete Hardy operator of the form , from to when 0 < q < 1 <p , is given.     

Boundedness and Asymptotic Behaviour of Solutions of a Second-Order Nonlinear System

Consider the second-order nonlinear system where a is a positive and continuous function onR:(– , ), b, c and h are continuous functions on R, ande is a continuous function on I:[0, ). We obtain a sufficientcondition for all solutions of (*) to be bounded, and obtaina sufficient condition for all solutions of (*) to tend to zero.Our results can be applied to the well-known equation which substantially improves and extendsseveral known results in the literature.     

Reducing Subspaces for a Class of Multiplication Operators

Let D be the open unit disk in the complex plane C. The Bergmanspace is the Hilbert space of analytic functions f in D such that where dA is the normalized area measure on D. If are two functions in , then the inner product of f and g is given by We study multiplication operators on induced by analytic functions. Thus for H (D), the space ofbounded analytic functions in D, we define by It is easy to check that M is a bounded linear operator on with ||M||=||||=sup{|(z)|:zD}.     

Estimates for the Norm of the nth Indefinite Integral

Let T be the Volterra operator on L²[0, 1] where f L²[0, 1], 0 x 1. It is well known that||n!Tⁿ|| = O(1/n!). In a recent paper [1], D. Kershaw has provedthat a result which wasfirst conjectured by Lao and Whitley in [2]. It is easy to provethat For completeness, wegive the proof using the familiar Schmidt norm estimate forthe norm of an integral operator (see Section 2 below). Kershawproves that by analysingthe special positivity preserving properties of T*T. He usesone of the many abstract theorems on eigenvalues and eigenfunctionsof compact operators which preserve a cone. In this paper weshall reprove (1), giving a short and direct proof of (2). 1991Mathematics Subject Classification 47G10, 45-04.     

Calculs De Facteurs Epsilon De Paires Pour GLn Sur Un Corps Local, I

Soient F un corps commutatif localement compact non archimédienet un caractère additif non trivial de F. Soient unereprésentation du groupe de Weil–Deligne de F,et sa contragrédiente. Nous calculons le facteur (, , ). De manière analogue, nous calculons le facteur (x, , ) pour toute représentationadmissible irréductible de GL_n(F). En conséquence,si F est de caractéristique nulle et si et se correspondentpar la correspondance de Langlands construite par M. Harris,ou celle construite par les auteurs, alors les facteurs (, , s) et (x, , s) sont égaux pour tout nombre complexe s. Let F be a non-Archimedean local field and a non-trivial additivecharacter of F. Let be a representation of the Weil–Delignegroup of F and its contragredient representation. We compute (, , ). Analogously, we compute (x, , ) for all irreducible admissible representations of GL_n(F).Consequently, if F has characteristic zero, and , correspondvia the Langlands correspondence established by M. Harris orthe correspondence constructed by the authors, then we have(, , s) = (x, , s) for all sC. 1991 Mathematics Subject Classification22E50.     

A Local Limit Theorem for Moderate Deviations

The purpose of this note is to establish a uniform estimatefor the mass function P(S_m = y) of an integer-valued randomwalk when y and where µ isthe mean of the step distribution. (The local central limittheorem provides such an estimate when ( is bounded.) The assumptions are that the mass functionp of the step distribution is regularly varying at with index–, where > 3, and that for some ' > 2. From this result, a ratio limit theorem isderived, and this in turn is applied to yield some new informationabout the space–time Martin boundary of certain randomwalks.