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

Year 1977, Volume: 26 , 0 - 0, 01.01.1977

Gupta J.s.

https://doi.org/10.1501/Commua1_0000000275

Abstract

Letf(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) (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 !,(, 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 (C is the complex G is f (s) =
S a„ e®\ı with lim sup log n
nsN O' + «
= D s R_^ U {0} (R^ is the set of
•n
positive reals), and af c = + 05 (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 (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

References

• Ankara Üniversitesi – Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Dergisi