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Extra info for Distribution of Zeros of Entire Functions (Translations of Mathematical Monographs)
Example text
R) (for example. (r) is logarithmically convex). Sec. ;;; k < I. 52), for arbitrary 1] > 0 and all sufficiently large values of r we have by Lagrange's theorem: Ip(r) - p (kr) I < ~ ~ -<: '1l 1j ;,. (+ - 1) In :ar) . ;;; 1 · InL -(kr) I1m -= 0 f'-+OO L (r) L (kr) or lim -L() =1. ,. f'-+OO The case k > I is treated similarly. We now establish some simple properties of proximate orders that we shall need frequently. (I) For p > 0 the function r"'-r) is monotone increasing for all suffiCiently large values of r.
3. to this case. concave function y - 'Pl(X) so that lim ~l (x) a: -+ 00 = 0, lim 1\I~(x) = 0 a:-+oo· X and so that lim sUPz--co[9'l{X) + 'Pl(X)] = 00. ¥ from the origin to the point Xl (Fig. 2) at which CP1 (x t ) Choose a positive number EI t1a of the line >- '1 X 1 + 1. < El and from the point (Xl' -E1Xl ) pass a segment Y+11 Xt = - li (X-X1) to a point XI > Xl' at which 38 GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS [CHAP. ) etc. y r, -R~~T---~~------~-~ o r : \y=rp,(r) '--"-~~-~--y=~(r) Fig.
38') We put ~ _1- ~ aP lanl