On the Minimum Modulus of Analytic Functions of Moderate Growth in the Unit Disc
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On the Minimum Modulus of Analytic Functions of Moderate Growth in the Unit Disc I. Chyzhykov1,2 · M. Kravets1
Received: 25 July 2014 / Revised: 19 September 2014 / Accepted: 10 March 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract We study the behavior of the minimum modulus of analytic functions in the unit disc in terms of ρ∞ -order, which is the limit of the orders of L p -norms of log | f (r eiθ )| over the circle as p → ∞. This concept coincides with the usual order of the maximum modulus function if the order is greater than one. New results are obtained for analytic functions of order smaller than 1. Keywords Analytic function · Minimum modulus · Order of growth · Factorization · Zero distribution · Canonical product · Harmonic function Mathematics Subject Classification
30J99 · 30H05 · 30H15 · 30J10
1 Introduction and Main Results Let D R = {z ∈ C : |z| < R}, 0 < R ≤ ∞, and D = D1 . For an analytic function f on D R , we define the minimum modulus μ(r, f ) = min{| f (z)| : |z| = r }, 0 < r < R,
Communicated by James K. Langley.
B
I. Chyzhykov [email protected] M. Kravets [email protected]
1
Faculty of Mechanics and Mathematics, Ivan Franko National University of Lviv, Universytets’ka 1, Lviv 79000, Ukraine
2
Present Address: Faculty of Mathematics and Natural Studies, Cardinal Stefan Wyszy´nski University in Warsaw, Wóycickiego 1/3, 01-938 Warszawa, Poland
123
I. Chyzhykov, M. Kravets
and the maximum modulus M(r, f ) = max{| f (z)| : |z| = r }, 0 < r < R. Interplay between μ(r, f ) and M(r, f ) has been studied in a large number of papers. In the case of entire functions, i.e., R = ∞, a survey of results up to 1989 can be found in Hayman’s book ([16, Chap. 6]). The orders of the growth of an analytic function f in D∞ , and in D, respectively, are defined as ρ[ f ] = lim sup r ∞
log+ log+ M(r, f ) , log r
ρ M [ f ] = lim sup r 1
log+ log+ M(r, f ) . − log(1 − r )
For entire functions of order ρ[ f ] ≤ 1, there are a lot of sharp results on the behavior such as cos πρ-theorem ([1,16]). Theorem ([1]) Suppose that 0 ≤ ρ < α < 1. If f is an entire function of order ρ and f (z) ≡ const then log μ(r, f ) ≥ cos π α log M(r, f ), r ∈ E where lim
r →∞
dt E∩[1,r ) t
log r
≥1−
ρ . α
One of the most interesting open problems for entire functions of order greater than 1 is to find the asymptotic behavior of the minimum modulus with respect to the maximum modulus, especially for values of ρ[ f ] close to 1 ([14,15]). The most precise results concerning the minimum modulus of entire and subharmonic functions of order zero can be found in [2–4,11–13]. For analytic functions in the unit disc D the situation, in a certain sense, is the opposite. Known results are much weaker in accuracy than the statements of the cos πρ-theorem type. Moreover, these results mainly concern analytic functions with ρ M [ f ] ≥ 1. We start with an old result of M. Heins. Theorem A [18] If f (z) is analytic in D, f (z) ≡ const, f (z) is bounded in D, then there ex
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