The Wiman-Valiron method for entire functions, defined by Dirichlet series, with a growth condition on certain sequences

Relative growth of terms of a Dirichlet series: the use of convexity.

“…It has to be said that our results are entirely new, s far s we know, for R < oo (in [11] the problem is only slightly touched for power series with gaps in a disk); in case R = oo, s /(z) in (l .1) is an entire transcendental function of finite order [4] or finite lower order [1], [2] our connections differ by certain constants from the relations obtained in the articles above -the latter being preferable (in [5] and [8] the relevant connections differ in form and content from the ones in [1], [3], [4] and in this work; see, by the way [9], where various known approaches to Wiman-Valiron type estimates are discussed). Besides the new results we prove in the present article, our mode of reasoning may be of interest in itself.…”

“…(1.5) \ημ(χ)-\ημ(χ 0 )= expressing the convexity of 1ημ(.χ) (see by the way [4], [5]), is the key tool we use in our forthcoming study.…”