Hadamard convexity and multiplicity and location of zeros

Abi-Khuzam, Faruk F.

doi:10.1090/S0002-9947-1995-1285968-9

Hadamard convexity and multiplicity and location of zeros
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by Faruk F. Abi-Khuzam PDF

Trans. Amer. Math. Soc. 347 (1995), 3043-3051 Request permission

Abstract:

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

References

Faruk F. Abi-Khuzam, Maximum modulus convexity and the location of zeros of an entire function, Proc. Amer. Math. Soc. 106 (1989), no. 4, 1063–1068. MR 972225, DOI 10.1090/S0002-9939-1989-0972225-3
V. S. Boĭčuk and A. A. Gol′dberg, On the three lines theorem, Mat. Zametki 15 (1974), 45–53 (Russian). MR 344465
W. K. Hayman, Note on Hadamard’s convexity theorem, Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966) Amer. Math. Soc., Providence, R.I., 1968, pp. 210–213. MR 0252639

The convexity theorem of Hadamard-Hayman

Review of

E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759