K-th Mean Function of Entire Functions Defined by Dirichlet Series

Authors : Gupta JS

Pages : 0-0

Doi:10.1501/Commua1_0000000275

View : 17 | Download : 11

Publication Date : 1977-01-01

Article Type : Research Paper

Abstract :Letfinsert ignore into journalissuearticles values(s); = S “sN % be an entire function defined by an everywbere convergent Dirichlet series whose exponents are subjected to the condition lim sup co loğu = Ds u |o); insert ignore into journalissuearticles values(R_|_ is the set of positive reals);. The notion of K-th mean function öf f was iuterduced by the first author in [2]. We generalize !insert ignore into journalissuearticles values(, and define r e R, as 1 7® .rx dx, Vas: R, and study some propertes of and 0` Ijj j, in tbis paper. Beside establisbing the convexity of we have derived some formulas for Ritt order and lovver order of f in terms of and which are improvements and generalizations of known ones. AMS subject classification number: Primary 30A64 Secondary 30A62. Key Words: Entire function, Dirichlet series, manmnm modulus, mavimum term, rank, K-th mean function, convex function, Ritt order, lower order. 1. Let E be the set of mappings f: C field); such that the image under f of an element s s C insert ignore into journalissuearticles values(C is the complex G is f insert ignore into journalissuearticles values(s); = S a„ e®\ı with lim sup log n nsN O` + « = D s R_^ U {0} insert ignore into journalissuearticles values(R^ is the set of •n positive reals);, and af c = + 05 insert ignore into journalissuearticles values(cf c is the absissa of convergen- ce of the Dirichlet series defining f);; N is the set of natural num- bers 0, 1, 2, ..,„ | seguence of uonnegative reals, s = n e N> is a strictly increasing unbounded q + it^ c, t e R insert ignore into journalissuearticles values(R İs the field of reals);, and is a seguence in C. Since the Dirich- let series defining f converges for each complex s, f is an entire funtion.
Keywords : K th Mean Function, Entire Function, Dirichlet Series