Subharmonic Functions, Generalizations, Holomorphic Functions, Meromorphic Functions, and Properties

2021 ◽  
Author(s):  
Juhani Riihentaus
Keyword(s):  
Meromorphic Functions ◽  
Holomorphic Functions ◽  
Subharmonic Functions

On Classes for Hyperbolic Riemann Surfaces

2016 ◽  
Vol 59 (01) ◽  
pp. 13-29
Author(s):  
Rauno Aulaskari ◽  
Huaihui Chen
Keyword(s):  
Unit Ball ◽  
Riemann Surfaces ◽  
Meromorphic Functions ◽  
Holomorphic Functions ◽  
Spaces Of Holomorphic Functions ◽  
Inclusion Relations ◽  
Hyperbolic Riemann Surfaces ◽  
Qp Spaces

AbstractThe Qpspaces of holomorphic functions on the disk, hyperbolic Riemann surfaces or complex unit ball have been studied deeply. Meanwhile, there are a lot of papers devoted to theclasses of meromorphic functions on the disk or hyperbolic Riemann surfaces. In this paper, we prove the nesting property (inclusion relations) ofclasses on hyperbolic Riemann surfaces. The same property for Qp spaces was also established systematically and precisely in earlier work by the authors of this paper.

The lemma of the logarithmic derivative for subharmonic functions

1996 ◽  
Vol 120 (2) ◽  
pp. 347-354 ◽  
Author(s):  
Walter Rudin
Keyword(s):  
Lebesgue Measure ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Value Distribution ◽  
Nevanlinna Characteristic ◽  
Value Distribution Theory ◽  
Subharmonic Functions ◽  
Classical Statement ◽  
Image Position ◽  
Finite Lebesgue Measure

The classical statement of the lemma in question [7], [3] is about meromorphic functions f on ℂ and says thatfor all r > 0, with the possible exception of a set of finite Lebesgue measure. Here T(r, f) is the Nevanlinna characteristic of f. The lemma plays an important role in value distribution theory.

scholarly journals A normal criterion concerning zero numbers

2021 ◽  
Author(s):  
Chengxiong Sun
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Holomorphic Functions ◽  
Normal Criterion

AbstractLet $$n \ge 4$$ n ≥ 4 be a positive integer, $$\mathcal {F}$$ F be a family of meromorphic functions in D and let $$a(z)(\not \equiv 0), b(z)$$ a ( z ) ( ≢ 0 ) , b ( z ) be two holomorphic functions in D. If, for any function $$f \in \mathcal { F}$$ f ∈ F , (1)$$f(z) \ne \infty $$ f ( z ) ≠ ∞ when $$a(z)=0$$ a ( z ) = 0 , (2) $$f'(z)-a(z)f^{n}(z)-b(z)$$ f ′ ( z ) - a ( z ) f n ( z ) - b ( z ) has at most one zero in D, then $$\mathcal {F}$$ F is normal in D.

Exceptional functions and normal families of holomorphic functions with multiple zeros

2011 ◽  
Vol 18 (1) ◽  
pp. 31-38
Author(s):  
Jun-Fan Chen
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Normal Family ◽  
Holomorphic Functions ◽  
Normal Families ◽  
Multiple Zeros ◽  
Zeros Of Functions

Abstract Let k be a positive integer, and let ℱ be a family of functions holomorphic on a domain D in C, all of whose zeros are of multiplicity at least k + 1. Let h be a function meromorphic on D, h ≢ 0, ∞. Suppose that for each ƒ ∈ ℱ, ƒ(k)(z) ≠ h(z) for z ∈ D. Then ℱ is a normal family on D. The condition that the zeros of functions in ℱ are of multiplicity at least k + 1 cannot be weakened, and the corresponding result for families of meromorphic functions is no longer true.

scholarly journals Norm estimates for functions of non-selfadjoint operators nonregular on the convex hull of the spectrum

2017 ◽  
Vol 50 (1) ◽  
pp. 267-277 ◽  
Author(s):  
Michael Gil’
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Convex Hull ◽  
Meromorphic Functions ◽  
Holomorphic Functions ◽  
Fractional Powers ◽  
Bounded Linear ◽  
Norm Estimates ◽  
Selfadjoint Operators ◽  
Finite Dimensional

Abstract We consider a bounded linear operator A in a Hilbert space with a Hilbert-Schmidt Hermitian component (A − A*)/2i. A sharp norm estimate is established for functions of A nonregular on the convex hull of the spectrum. The logarithm, fractional powers and meromorphic functions of operators are examples of such functions. Our results are based on the existence of a sequence An (n = 1, 2, ...) of finite dimensional operators strongly converging to A, whose spectra belongs to the spectrum of A. Besides, it is shown that the resolvents and holomorphic functions of An strongly converge to the resolvent and corresponding function of A.

scholarly journals Blaschke-type maps and harmonic majoration on Riemann surfaces

1985 ◽  
Vol 32 (2) ◽  
pp. 195-205
Author(s):  
Shinji Yamashita
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Holomorphic Functions ◽  
Subharmonic Functions

An analytic map h of type Bl from a Riemann surface R into another S, both having Greens functions, behaves well near the “boundaryr” of R. Let X stand for a family of holomorphic functions, and let f be holomorphic on S. We shall show, for several X′s, the following:(i) f ∈ X(S) ⇔ foh ⇔ X(R);‖foh‖ = ‖f‖.Use is made of harmonic majoration of subharmonic functions on R and on S.

Fixed points of composite meromorphic functions and normal families

2004 ◽  
Vol 134 (4) ◽  
pp. 653-660 ◽  
Author(s):  
Walter Bergweiler
Keyword(s):  
Fixed Point ◽  
Meromorphic Function ◽  
Fixed Points ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Holomorphic Functions ◽  
Normal Families ◽  
The Family

We show that there exists a function f, meromorphic in the plane C, such that the family of all functions g holomorphic in the unit disc D for which f ∘ g has no fixed point in D is not normal. This answers a question of Hinchliffe, who had shown that this family is normal if Ĉ\f(C) does not consist of exactly one point in D. We also investigate the normality of the family of all holomorphic functions g such that f(g(z)) ≠ h(z) for some non-constant meromorphic function h.

scholarly journals Some unsolved problems on meromorphic functions of uniformly bounded characteristic

1985 ◽  
Vol 8 (3) ◽  
pp. 477-482 ◽  
Author(s):  
Shinji Yamashita
Keyword(s):  
Characteristic Function ◽  
Meromorphic Functions ◽  
Holomorphic Functions ◽  
Bounded Mean Oscillation ◽  
Open Problems ◽  
Mean Oscillation ◽  
The Family ◽  
Bounded Characteristic ◽  
Uniformly Bounded

The familyUBC(R)of meromorphic functions of uniformly bounded characteristic in a Rieman surfaceRis defined in terms of the Shimizu-Ahlfors characteristic function. There are some natural parallels betweenUBC(R)andBMOA(R), the family of holomorphic functions of bounded mean oscillation inR. After a survey some open problems are proposed in contrast withBMOA(R).

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