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$\cos\pi\lambda$ again

Author(s): P. C. Fenton
Journal: Proc. Amer. Math. Soc. 131 (2003), 1875-1880.
MSC (2000): Primary 30D15
Posted: November 6, 2002
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Abstract: It is shown that if, for an entire function,

$\begin{displaymath}\liminf_{r\to\infty} \logEM(r)/r^{\lambda} = 0 \end{displaymath}$

where $0< \lambda <1$ , then

$\begin{displaymath}\limsup_{r\to\infty}(\log m(r)-\cos\pi\lambda\log M(r))/\log r = \infty. \end{displaymath}$

In the proof, the zeros of the function are redistributed to minimize the large values of $\log m(r) -\cos\pi\lambda\log M(r)$ .

References:

1.: Albert Baernstein II, A generalization of the cos $\pi\rho$ theorem, Trans. Amer. Math. Soc. 193 (1974), 181-97. MR 49:9207
2.: R.P. Boas, Entire Functions (Academic Press, 1954). MR 16:914f
3.: E.T. Copson, Theory of Functions of a Complex Variable (Oxford, 1935).
4.: P.C. Fenton, A min-max theorem for sums of translates of a function, J. Math. Anal. App. 244 (2000), 214-222. MR 2001a:26007
5.: Bo Kjellberg, On the minimum modulus of entire functions of lower order less than one, Math. Scand. 8 (1960), 189-97. MR 23:A3264
6.: Bo Kjellberg, A theorem on the minimum modulus of entire functions, Math. Scand. 12 (1963), 5-11. MR 28:3158


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