##
**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 N_{k}(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 **lim**_{k ––> +oo} k^{-1} N_{k}(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 **liminf**_{r ––> +oo} **{**g(r)**}**^{-1} log M(r) < 1, we have

**liminf**_{k ––> +oo} k^{-1} N_{k}(f(z),1) h(k) \leq e^{2}. Theorem 3. If f(z) is an integral function and z_{k} 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 **liminf**_{k ––> +oo} **{**k|z_{k}|**}**^{-1} H(k) \leq e(log 2)^{-1}.

Theorem 4. If f(z) is regular in |z| < R and is not a polynomial, then **limsup**_{k ––> +oo} k|z_{k}| \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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag