## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  070.29601
Autor:  Erdös, Pál; Rényi, Alfréd
Title:  On the number of zeros of successive derivatives of analytic functions. (In English. RU summary)
Source:  Acta Math. Acad. Sci. Hung. 7, 125-144 (1956).
Review:  Let Nk(f(z),r) denote the number of zeros of f(k)(z) in |z| \leq r < R. The following theorems, which include and extend results of G.Pólya [Bull. Am. Math. Soc. 49, 178-191 (1943; Zbl 061.11510)] and Evgrafov (Interpolationsaufgabe von Abel-Goncarov, Moskau 1954) are proved.
Theorem 1. If f(z) is regular in |z| < 1 and 0 < r < 1, then limk ––> +oo k-1 Nk(f(z),r) \leq K(r), where K(r) is the only positive root of K = r(1+K)1+1/K.
Theorem 2. Let g(r) \uparrow+oo in 0 < r <+oo. Let x = h(y) denote the inverse function of y = g(x). Then, if f(z) is an integral function which satisfies

liminfr ––> +oo {g(r)}-1 log M(r) < 1,

we have

liminfk ––> +oo k-1 Nk(f(z),1) h(k) \leq e2.

Theorem 3. If f(z) is an integral function and zk is the zero of f(k)(z) which is nearest the origin (k = 1,2,3,...), and if x = H(y) is the inverse function of y = log M(x), then liminfk ––> +oo {k|zk|}-1 H(k) \leq e(log 2)-1.
Theorem 4. If f(z) is regular in |z| < R and is not a polynomial, then limsupk ––> +oo k|zk| \geq R log 2. The proof are based on Jensen's formula and Rouché's theorem.
Reviewer:  N.A.Bowen
Classif.:  * 30C15 Zeros of polynomials, etc. (one complex variable)
Index Words:  complex functions

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