On The Mean Values of an Entire Function Represented By a Dirichlet Series II

Authors : Jodh Pal SINGH

Pages : 0-0

Doi:10.1501/Commua1_0000000636

View : 36 | Download : 7

Publication Date : 1973-01-01

Article Type : Research Paper

Abstract :In this note we prove a theorem which gives us Information as to how the functions log I§ insert ignore into journalissuearticles values(a); and log insert ignore into journalissuearticles values(a); grow relative to each other as a ^00. Theorem. Let finsert ignore into journalissuearticles values(s); be an entire function represented by a Dirichlet series, then lim a ^00 log Ig insert ignore into journalissuearticles values(<’); log J8,k insert ignore into journalissuearticles values(o); X ■! insert ignore into journalissuearticles values(1 + k/X); insert ignore into journalissuearticles values(X 0); insert ignore into journalissuearticles values(X = 0); where X = lim CT-^OO log log Ig insert ignore into journalissuearticles values(a); a 1. In the usual notation, 00 finsert ignore into journalissuearticles values(s); = s 1 Sn e®^-”, insert ignore into journalissuearticles values(s = insert ignore into journalissuearticles values(j it);, 0 A, insert ignore into journalissuearticles values(n 1); lim n -*oo 00 is an entire function in the sense that the Dirichlet series represen- ting it, is ahsolutely convergent for ali finite s and possesses where 0 lim insert ignore into journalissuearticles values(7-4-00 log log Minsert ignore into journalissuearticles values(g); insert ignore into journalissuearticles values(7 X, 00, and Minsert ignore into journalissuearticles values(ct); have their usual meanings.
Keywords : Mean Values, Entire Function Represented, Dirichlet Series

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