Hadamard convexity and multiplicity and location of zeros

Abstract

We consider certain questions arising from a paper of Hayman concerning quantitative versions of the Hadamard three-circle theorem for entire functions. If b ( r ) b(r) denotes the second derivative of log ⁡ M ( r ) \log M(r) with respect to log ⁡ r \log r , the principal contributions of this work are (i) a characterization of those entire f f with nonnegative Maclaurin coefficients for which lim sup b ( r ) = 1 4 \lim \sup b(r) = \frac {1} {4} and (ii) some exploration of the relationship between multiple zeros of f f and the growth of b ( r ) b(r) .