The Average Norm of Polynomials of Fixed Height
نویسندگان
چکیده
Let n ≥ 0 be any integer and let Fn := { n ∑ i=0 aiz i : ai = 0,±1 } be the set of all polynomials of height 1 and degree n. Let βn(m) := 1 3n+1 ∑ P∈Fn ‖P‖m. Here ‖P‖m is the mth power of the Lm norm on the boundary of the unit disc. So βn(m) is the average of the mth power of the Lm norm over Fn. In this paper we give exact formulae for βn(m) for various values of m. We also give a variety of related results for different classes of polynomials including polynomials of fixed height H, polynomials with coefficients ±1 and reciprocal polynomials. The results are surprisingly precise. Typical of the results we get is the following. Theorem 0.1. For n ≥ 0, we have βn(2) = 2 3 (n+ 1), βn(4) = 8 9 n + 14 9 n+ 2 3 and βn(6) = 16 9 n + 4n + 26 9 n+ 2 3 .
منابع مشابه
ar X iv : m at h / 02 12 34 4 v 1 [ m at h . N T ] 2 6 D ec 2 00 2 AVERAGE NORMS OF POLYNOMIALS
In this paper we study the average L2α-norm over T -polynomials, where α is a positive integer. More precisely, we present an explicit formula for the average L2α-norm over all the polynomials of degree exactly n with coefficients in T , where T is a finite set of complex numbers, α is a positive integer, and n ≥ 0. In particular, we give a complete answer for the cases of Littlewood polynomial...
متن کاملThe norm of bioenergetics characteristics of elite and advanced male and female Iranian Badminton players
The purpose of this study was to collect the norm of bio-energetic characteristics of male and female elite and advanced Badminton players in Iran based on gender differences. This is a descriptive research of the kind of normative survey. Statistical population of the research includes all elite and advanced male and female Badminton players in Iran during 2010 including 128 elite and adv...
متن کاملAn equivalent representation for weighted supremum norm on the upper half-plane
In this paper, rstly, we obtain some inequalities which estimates complex polynomials on the circles.Then, we use these estimates and a Moebius transformation to obtain the dual of this estimates forthe lines in upper half-plane. Finally, for an increasing weight on the upper half-plane withcertain properties and holomorphic functions f on the upper half-plane we obtain an equivalentrepresenta...
متن کاملChromatic polynomials of some nanostars
Let G be a simple graph and (G,) denotes the number of proper vertex colourings of G with at most colours, which is for a fixed graph G , a polynomial in , which is called the chromatic polynomial of G . Using the chromatic polynomial of some specific graphs, we obtain the chromatic polynomials of some nanostars.
متن کامل