Estimates for the volume of zeros of a holomorphic function depending on a complex parameter

2021 ◽  
Vol 212 (11) ◽  
Author(s):  
Aleksandr Mechislavovich Kytmanov ◽  
Azimbay Sadullaevich Sadullaev
Keyword(s):  
Holomorphic Function ◽  
Complex Parameter

The Complex Parameter Rao Test

2016 ◽  
Vol 64 (24) ◽  
pp. 6580-6588 ◽  
Author(s):  
Steven Kay ◽  
Zhenghan Zhu
Keyword(s):  
Complex Parameter ◽  
Rao Test

scholarly journals A Criterion for Normalcy

1968 ◽  
Vol 32 ◽  
pp. 277-282 ◽  
Author(s):  
Paul Gauthier
Keyword(s):  
Meromorphic Function ◽  
Holomorphic Function ◽  
Unit Disc ◽  
Meromorphic Functions

Gavrilov [2] has shown that a holomorphic function f(z) in the unit disc |z|<1 is normal, in the sense of Lehto and Virtanen [5, p. 86], if and only if f(z) does not possess a sequence of ρ-points in the sense of Lange [4]. Gavrilov has also obtained an analagous result for meromorphic functions by introducing the property that a meromorphic function in the unit disc have a sequence of P-points. He has shown that a meromorphic function in the unit disc is normal if and only if it does not possess a sequence of P-points.

A Characterization of Bergman Spaces on the Unit Ball of ℂn. II

2012 ◽  
Vol 55 (1) ◽  
pp. 146-152 ◽  
Author(s):  
Songxiao Li ◽  
Hasi Wulan ◽  
Kehe Zhu
Keyword(s):  
Holomorphic Function ◽  
Unit Ball ◽  
Bergman Space ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Double Integrals

AbstractIt has been shown that a holomorphic function f in the unit ball of ℂn belongs to the weighted Bergman space , p > n + 1 + α, if and only if the function | f(z) – f(w)|/|1 – 〈z, w〉| is in Lp( × , dvβ × dvβ), where β = (p + α – n – 1)/2 and dvβ(z) = (1 – |z|2)βdv(z). In this paper we consider the range 0 < p < n + 1 + α and show that in this case, f ∈ (i) if and only if the function | f(z) – f(w)|/|1 – hz, wi| is in Lp( × , dvα × dvα), (ii) if and only if the function | f(z)– f(w)|/|z–w| is in Lp( × , dvα × dvα). We think the revealed difference in the weights for the double integrals between the cases 0 < p < n + 1 + α and p > n + 1 + α is particularly interesting.

scholarly journals Generalised sturm-liouville expansions in series of pairs of related functions

1927 ◽  
Vol 115 (770) ◽  
pp. 184-198 ◽  
Keyword(s):  
Boundary Conditions ◽  
Differentiable Function ◽  
Real Variable ◽  
Complex Parameter ◽  
The Real ◽  
The Earth ◽  
In Series ◽  
Sturm Liouville

The expansions here developed are required for the author’s discussion of "Meteorological Perturbations of Tides and Currents in an Unlimited Channel rotating with the Earth” ( v. supra , p. 170). Let η ( x ) be a real differentiable function of x defined in the range 0 ≼ x ≼ 1, and satisfying the condition η ( x ) > c > 0 for all such x . Let ϕ λ ( x ) and ψ λ ( x ) be functions of the real variable x and the complex parameter λ , defined in the above range by the equations d / dx [ η ( x ) dϕ λ ( x )/ dx ] + ( λ + iγ ) ϕ λ ( x ) = -1, d / dx [ η ( x ) dψ λ ( x )/ dx ] + ( λ + iγ ) ψ λ ( x ) = -1 (1) together with the boundary conditions ϕ' λ (0) = 0, ψ' λ (0) = 0, ϕ' λ (1) = 0, ψ λ (1) = 0, (2) γ being a prescribed constant.

scholarly journals Complex-Parameter Integral Iterations of Caratheodory Maps

2020 ◽  
Vol 51 (4) ◽  
pp. 289-301
Author(s):  
Kunle Oladeji Babalola ◽  
Mashood Sidiq
Keyword(s):  
Complex Parameter

Recent studies in the class of Bazilevi$\check{c}$ maps as a whole has compelled the development, in this work, of certain complex-parameter integral iterations of Caratheodory maps. The iterations are employed in a similar manner as in \cite{BA} to study a certain subfamily of those Bazilevi$\check{c}$ maps.

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