Identity Theorems for Functions of Bounded Characteristic

Suppose that f ( z ) is a meromorphic function of bounded characteristic in the unit disk Δ:∣ z ∣<1. Then we shall say that f ( z )∈ N . It follows (for example from [ 3 , Lemma 6.7, p. 174 and the following]) that f ( z ) = c∏ 1 ( z )∏ 2 ( z )e h 1 ( z ) ‐ h 2 ( z )where h 1 ( z ), h 2 ( z ) are holomorphic in Δ and have positive real part there, while Π 1 ( z ), Π 2 ( z ) are Blaschke products, that is,∏ 1 ( z ) = z p∏ j {z ‐ a j1 ‐a ¯ j z} . {‐a j ¯| a j |}where p is a positive integer or zero, 0<∣ a j ∣<1, c is a constant and ∑(1−| a j |)<∞. We note in particular that, if c ≠0, so that f ( z )≢0, 1.1| f ( z ) | ⩾ | c ∏ 1 ( z ) e ‐ h 2 ( z )| ⩾ | c ∏ 1 ( z ) | e ‐ | h 2 ( 0 ) |( 1 + | z | ) / ( 1 ‐ | z | )so that f ( z )=0 only at the points a j . Suppose now that z j is a sequence of distinct points in Δ such that | z j |→1 as j →∞ and ∑(1−| z j |)=∞. (1.2) If f ( z j )=0 for each j and f ∈ N , then f ( z )≡0. N. Danikas [ 1 ] has shown that the same conclusion obtains if f ( z j )→0 sufficiently rapidly as j →∞. Let ε j , λ j be sequences of positive numbers such that ∑ε j <∞ and λ j →∞ as j →∞. Danikas then definesδ j = min { ε j , 1 2inf k ≠ j| z k ‐ z j | }and proves Theorem A.