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.