Laplace-Stieltjes变换所定义的函数在收敛半平面内的增长性

Laplace-Stieltjes变换所定义的函数在收敛半平面内的增长性 THE GROWTH OF A FUNCTION DEFINED BY A LAPLACE-STIELTJES TRANSFORM IN ITS HALF PLANE OF CGNVERGENCE 李至琳 Li Zhilin 1 first-author 黔东南民族师专 黔东南民族师专 <正> Mandelbrojt S.[3],Blambert M.[2]和余家荣[6]等人研究过Dirichlet级数所确定整函数的Ritt级和型与级数系数之间的关系。余家荣[8]定义Dirichlet级数所确定函数在收敛半平面内的级和准确级,导出了Dirichlet级数的增长性与其系数之间的一些关系,这些结果包含并推广了Valiron G.[4]关于Taylor级数的结果。余家荣[7]在研究Laplace-Stieltjes变换所定义的整函数的Borel线时,引进了Laplace-Stieltjes变换所定义的整函数之"最大模"与"最大项"的定义,导出了它们的关系式,从而对Laplace-Stieltjes变换所定义的整函数建立了Liouville型的定理。本文按照文献[7],将Dirichlet级数在收敛半平面内的增长性的主要结果[8],相应地转移到Laplace-Stieltjes变换所定义的函数上。考虑Laplace-Stieltjes变换所定义的函数 Consider a function F（s） defined by a Laplace-Stieltjes transform where α（x） be a function of real or Complex Values defined on x≥0, and α（x） is of bounded variation in every interal [0, x]（0 < x < +∞）. Suppose0=λ 0 < λ 1 < λ 2 … < λ n ↑+∞, that(λ n+1 -λ n ) < +∞ and（logn）/（λ n ）=0. Let and If both the abscissa of convergence and the abscissa of absolute convergence of F（s）are zero, then（log A n * ）/（λ n ）=0, and We have the following inequalities and where K is a positive constant sueh that λ n+1 -λ n ≤K（n=0, 1, 2, …）. We define the Ritt order and Proximate order of F（s） in α > 0. So we obtain Some results on the growth of a function defined by a Laplace-Stieltjes transform in its half plane of Convergence, Corresponding those of professor Yu Jiarong for Dirichlet Series[8]. 1981-03-01 2021-04-01 3