COROLLARY 5.2. limH n 1 ' n = H. n-»ooProof. Let ji ^ 1 be fixed and write n = qj + r, 0 ^ r < j . Repeated applications of Theorem 5.1 yield H n > H qj H r > HfHf = Hfef.Since q = (n -r)/j, we have Letting n -> oo, we obtain »->ooSince this holds for every positive integer j , we have which completes the proof.

In what follows, we suppose that ƒ(z) = Σ0∞anzn is regular for |z| < 1. LetandThen (see, for example, [6, pp. 235-236]), for 0 ≦ r < ρ < 1, we have:The following results are well known.

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