Hadamard convexity and multiplicity and location of zeros

Author:
Faruk F. Abi-Khuzam

Journal:
Trans. Amer. Math. Soc. **347** (1995), 3043-3051

MSC:
Primary 30D20; Secondary 30D15, 30D35

DOI:
https://doi.org/10.1090/S0002-9947-1995-1285968-9

MathSciNet review:
1285968

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Abstract: We consider certain questions arising from a paper of Hayman concerning quantitative versions of the Hadamard three-circle theorem for entire functions. If denotes the second derivative of with respect to , the principal contributions of this work are (i) a characterization of those entire with nonnegative Maclaurin coefficients for which and (ii) some exploration of the relationship between multiple zeros of and the growth of .

**[1]**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**, https://doi.org/10.1090/S0002-9939-1989-0972225-3**[2]**V. S. Boĭčuk and A. A. Gol′dberg,*On the three lines theorem*, Mat. Zametki**15**(1974), 45–53 (Russian). MR**0344465****[3]**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****[4]**B. Kjelleberg,*The convexity theorem of Hadamard-Hayman*, Proc. Sympos. Math., Stockholm (June 1973, Royal Institute of Technology), pp. 87-114.**[5]**-,*Review of*, Zentralblatt für Math., 1990.**[6]**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**

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DOI:
https://doi.org/10.1090/S0002-9947-1995-1285968-9

Article copyright:
© Copyright 1995
American Mathematical Society