Vol. 54, No. 2, 1974

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Differential inequalities and local valency

Qazi Ibadur Rahman and Jan Stankiewicz

Vol. 54 (1974), No. 2, 165–181

An entire function f(z) is said Co have bounded value distribution (b.v.d.) if there exist constants p, R such that the equation f(z) = w never has more than p roots in any disk of radius R. It was shown by W. K. Hayman that this is the case for a particular p and some R > 0 if and only if there is a constant C > 0 such that for all z

|f (p+1)(z)| ≤ C max |f(ν)(z)|,

so that f(z) has bounded index in the sense of Lepson.

The fact that f(z) has bounded index if f(z) has b.v.d. follows readily from a classical result on p-valent functions. In the other direction Hayman proved that if

 (n)              (ν)
|f   (z)| ≤ 0≤mνa≤xn−1|f (z)|,

then f(z) cannot have more than n 1 zeros in |z|≤√--
n∕e√ --
20. Here the order of magnitude is correct in the sense that √ --
20 cannot be replaced by √-
3√ --
n. The result when applied to f(z) w does show that f(z) has bounded index only iff(z) has b.v.4. but it is clearly of interest to determine the largest disk containing at most n 1 zeros of f(z). We are able to replace √--
n∕e√ --
20 by √ --

Mathematical Subject Classification
Primary: 30A04
Received: 1 June 1972
Published: 1 October 1974
Qazi Ibadur Rahman
Jan Stankiewicz