Entire Functions with Bounded Minimum Modulus; Subharmonic Function Analogues

1948 ◽  
Vol 49 (1) ◽  
pp. 200 ◽  
Author(s):  
Maurice Heins
Keyword(s):  
Subharmonic Function ◽  
Entire Functions ◽  
Minimum Modulus

On the Growth of Entire Functions Bounded on Large Sets

1977 ◽  
Vol 29 (6) ◽  
pp. 1287-1291
Author(s):  
Lowell J. Hansen
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Maximum Modulus ◽  
Modulus Function ◽  
Classical Theorem ◽  
Minimum Modulus ◽  
Rate Of Growth ◽  
Large Sets ◽  
Growth Of Entire Functions ◽  
Lindelöf Theorem

There have been many indications of a relationship between the rate of growth of an entire function and the “size” of the set, E(c), where the modulus of the function is larger than the constant, c. Theorems of this type include the classical theorem of Wiman on functions of bounded minimum modulus, the Phragmén-Lindelöf Theorem, the Denjoy-Carleman-Ahlfors Theorem, and its many subsequent improvements. These theorems can all be understood as quantitative versions of the statement that if ƒ is an entire function such that, for some c > 0, the set E(c) is ‘'small”, then the maximum modulus function M(R, f) is forced to grow rapidly with R.

scholarly journals Asymptotics of $\delta$-subharmonic functions of finite order

2020 ◽  
Vol 54 (2) ◽  
pp. 188-192
Author(s):  
M.V. Zabolotskyi
Keyword(s):  
Finite Order ◽  
Subharmonic Function ◽  
Entire Functions ◽  
Proximate Order ◽  
Subharmonic Functions ◽  
Riesz Measure

For $\delta$-subharmonic in $\mathbb{R}^m$, $m\geq2$, function $u=u_1-u_2$ of finite positiveorder we found the asymptotical representation of the form\[u(x)=-I(x,u_1)+I(x,u_2) +O\left(V(|x|)\right),\ x\to\infty,\]where $I(x,u_i)=\int\limits_{|a-x|\leq|x|}K(x,a)d\mu_i(a)$, $K(x,a)=\ln\frac{|x|}{|x-a|}$ for $m=2$,$K(x,a)=|x-a|^{2-m}-|x|^{2-m}$ for $m\geq3,$$\mu_i$ is a Riesz measure of the subharmonic function $u_i,$ $V(r)=r^{\rho(r)},$ $\rho(r)$ is a proximate order of $u$.The obtained result generalizes one theorem of I.F. Krasichkov for entire functions.

scholarly journals Iterating the Minimum Modulus: Functions of Order Half, Minimal Type

2021 ◽  
Author(s):  
D. A. Nicks ◽  
P. J. Rippon ◽  
G. M. Stallard
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Maximum Modulus ◽  
Regular Growth ◽  
Transcendental Entire Function ◽  
Minimum Modulus ◽  
Precise Control ◽  
Significant Rate ◽  
Small Growth ◽  
Minimal Type

AbstractFor a transcendental entire function f, the property that there exists $$r>0$$ r > 0 such that $$m^n(r)\rightarrow \infty $$ m n ( r ) → ∞ as $$n\rightarrow \infty $$ n → ∞ , where $$m(r)=\min \{|f(z)|:|z|=r\}$$ m ( r ) = min { | f ( z ) | : | z | = r } , is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order 1/2 minimal type if the maximum modulus of f has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg’s method of constructing entire functions of small growth, which allows rather precise control of m(r).

scholarly journals On Subharmonic and Entire Functions of Small Order: After Kjellberg

2021 ◽  
Author(s):  
Philip J Rippon ◽  
Gwyneth M Stallard
Keyword(s):  
Entire Functions ◽  
Sharp Estimate ◽  
Minimum Modulus ◽  
Multiply Connected Domain ◽  
Positive Harmonic Function ◽  
Precise Control ◽  
Multiply Connected ◽  
Novel Technique ◽  
Transcendental Entire Functions ◽  
General Method

Abstract We give a general method for constructing examples of transcendental entire functions of given small order, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. Our method involves a novel technique to obtain an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, based on comparing the Harnack metric and hyperbolic metric, which gives a sharp estimate for the growth in many cases. Dedicated to the memory of Paddy Barry.

scholarly journals Entire Functions of Bounded L-Index: Its Zeros and Behavior of Partial Logarithmic Derivatives

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Maximum Modulus ◽  
Minimum Modulus ◽  
Distribution Of Zeros ◽  
And Behavior ◽  
Logarithmic Derivatives

In this paper, we obtain new sufficient conditions of boundedness of L-index in joint variables for entire function in Cn functions. They give an estimate of maximum modulus of an entire function by its minimum modulus on a skeleton in a polydisc and describe the behavior of all partial logarithmic derivatives and the distribution of zeros. In some sense, the obtained results are new for entire functions of bounded index and l-index in C too. They generalize known results of Fricke, Sheremeta, and Kuzyk.

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