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Some time ago Barry [Proc. London Math. Soc. 12 (1962), 445–495] established the right connection between the smallest and largest values of small subharmonic functions on certain circles about the origin. The behaviour of functions extremal for this connection is investigated.
It is shown that an entire function having k distinct entire asymptotic functions of order less than ¼ is of lower order ½k, mean type at least; further that if f is of lower order ½k, mean type, then its order is ½k.
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