URS' For Weierstrass Products without Exponential Factors

Abstract

Let $ \\cal W $ be the set of entire functions equal to a Weierstrass product of the form $ {f(x)= Ax^q\\lim_{r \ o \\infty} \\prod_{|a_j|\\leq r}{(1- \\fraca {x} {a_j})}} $ where the convergence is uniform in all bounded subsets of $ {\\shadC} $ , let $ \\cal V $ be the set of $ f\\in {\\cal W} $ such that $ {\\shadC} [\\,f]\\subset {\\cal W} $ , and let $ {\\cal H} $ be the $ {\\shadC} $ -algebra of entire functions satisfying $ { {\\lim_{r\ o \\infty } } ({\\ln M(r,f) / r})=0} $ . Then $ \\cal H $ is included in $ {\\cal V} $ and strictly contains the set of entire functions of genus zero, (which, itself, strictly contains the $ {\\shadC} $ -algebra of entire functions of order 𝜌 < 1). Let $ n, m\\in {\\shadN} ^* $ satisfy n > m S 3. Let $ a\\in {\\shadC}^* $ satisfies $ {a^n\ ot = \\fraca{n^n}{(m^m(n-m)^{n-m}})} $ and assume that for every ( n m m )-th root ξ of 1 different from m 1, a satisfies further $ {a^{n}\ eq (1+\\xi )^{n-m} (\\fraca{n^n}{((n-m)^{n-m}m^m}))} $ . Let P ( X ) = X n m aX m + 1 and let T n,m ( a ) be the set of its zeros. Then T n,m ( a ) has n distinct points and is a urs for $ {\\cal V} $ . In particular this applies to functions such as sin x and cos x .