Shamil Makhmutov - Academia.edu (original) (raw)
Papers by Shamil Makhmutov
Bulletin of The Australian Mathematical Society, Feb 1, 2009
Functions in the meromorphic Besov, Q p and related classes are characterized in terms of double ... more Functions in the meromorphic Besov, Q p and related classes are characterized in terms of double integrals of certain oscillation quantities involving chordal distances. Some of the results are analogous to the corresponding results in the analytic case.
Complex Variables and Elliptic Equations, Oct 20, 2009
ABSTRACT We study meromorphic functions on the complex plane with given growth of the spherical d... more ABSTRACT We study meromorphic functions on the complex plane with given growth of the spherical derivative. We find a criterion of Lohwater–Pommerenke type for non-weighted Yosida functions, and apply this result to obtain a version of Lappan's 5-point theorem. Further, a necessary condition for all solutions of algebraic differential equation of nth order to be weighted Yosida functions is found.
Computational Methods and Function Theory, Sep 20, 2007
The growth of the Dirichlet integral for functions belonging to Q p but not to the Dirichlet spac... more The growth of the Dirichlet integral for functions belonging to Q p but not to the Dirichlet space is considered. These results are shown to be sharp in a certain sense.
Doklady Mathematics, 2000
Complex Variables and Elliptic Equations, Sep 1, 2009
ABSTRACT Let : [0, 1) → (0, ∞) be an increasing function. A meromorphic function f in the unit di... more ABSTRACT Let : [0, 1) → (0, ∞) be an increasing function. A meromorphic function f in the unit disc is said to be -normal if its spherical derivative f #(z) |f′(z)|/(1 + |f(z)|2) satisfies f #(z) = ((|z|)) as |z| → 1−. This article is devoted to the study of meromorphic -normal functions. In particular, an analogue of the Lohwater–Pommerenke theorem and several equivalent characterizations for -normal functions are established under certain regularity conditions on .
Canadian mathematical bulletin, Mar 1, 2004
The Q p spaces coincide with the Bloch space for p > 1 and are subspaces of BMOA for 0 < p ≤ 1. W... more The Q p spaces coincide with the Bloch space for p > 1 and are subspaces of BMOA for 0 < p ≤ 1. We obtain lower and upper estimates for the essential norm of a composition operator from the Bloch space into Q p , in particular from the Bloch space into BMOA.
Proceedings of the Japan Academy. Series A, Mathematical sciences, 1996
1. Introduction. Let H(D) be the space of analytic functions on the unit disk D. Every holomorphi... more 1. Introduction. Let H(D) be the space of analytic functions on the unit disk D. Every holomorphic self-map o:D-+ D induces a linear composition operator C from H(D) into itself as follows: Cf-f q), whenever f H(D). In this paper we consider composition operators from the Bloch space to the spaces of analytic Besov functions B, 1 < p < oo.
Hokkaido Mathematical Journal, Feb 1, 1997
Compactness of composition operators from the Bloch space B into the analytic Besov spaces B_{p} ... more Compactness of composition operators from the Bloch space B into the analytic Besov spaces B_{p} is characterized by the behavior of the hyperbolic derivative of self-maps of the unit disk D .
Journal of Mathematical Sciences, 2021
Meromorphic functions with a given growth of a spherical derivative on the complex plane are desc... more Meromorphic functions with a given growth of a spherical derivative on the complex plane are described in terms of the relative location of a-points of functions. The result obtained allows one to construct an example of a meromorphic function in C with a slow growth of Nevanlinna characteristics and arbitrary growth of the spherical derivative. In addition, based on the universality property of the Riemann zeta-function, we estimate the growth of the spherical derivative of ζ(z).
Bulletin of The Australian Mathematical Society, Aug 1, 2000
We characterise the compact composition operators from any Mobius invariant Banach space to VMOA,... more We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.
arXiv (Cornell University), Mar 7, 2007
We discuss the value distribution of quasimeromorphic mappings in R n with given behavior in a ne... more We discuss the value distribution of quasimeromorphic mappings in R n with given behavior in a neighborhood of an essential singularity.
Tohoku Mathematical Journal, Dec 1, 2020
We establish pointwise growth estimates for the spherical derivative of solutions of the first or... more We establish pointwise growth estimates for the spherical derivative of solutions of the first order algebraic differential equations. A generalization of this result to higher order equations is also given. We discuss the related question of when for a given class X of meromorphic functions in the unit disc, defined by means of the spherical derivative, and m ∈ N \ {1}, f m ∈ X implies f ∈ X. An affirmative answer to this is given for example in the case of UBC, the α-normal functions with α ≥ 1 and certain (sufficiently large) Dirichlet type classes.
The New York Journal of Mathematics, 2011
Blaschke products are used to construct concrete examples of analytic functions with good integra... more Blaschke products are used to construct concrete examples of analytic functions with good integrability and bad behavior of spherical derivative. These examples are used to show that none of the classes M # p , 0 < p < ∞, is contained in the α-normal class N α when 0 < α < 2. This implies that M # p is in a sense a much larger class than Q # p .
Nihonkai mathematical journal, Dec 1, 1999
We study the cluster sets for analytic functions in the unit disk. Lindel\"of and Meier types the... more We study the cluster sets for analytic functions in the unit disk. Lindel\"of and Meier types theorems are proved for analytic cluster sets.
Journal of Mathematical Sciences, Aug 9, 2019
In this paper, we consider classes of analytical functions that map the unit disk into itself. Fu... more In this paper, we consider classes of analytical functions that map the unit disk into itself. Functions of these classes can be described in terms of hyperbolic derivative and hyperbolic metric. Under an appropriate choice of the corresponding metrics, these classes are metric spaces. Functions of the hyperbolic classes considered generate composition operators from the Bloch space into classical spaces of analytical functions in the unit disk.
Complex Variables, Feb 1, 2001
ABSTRACT α-normal functions, α ≥ 1, are meromorphic functions in the unit disk D with α-normal fu... more ABSTRACT α-normal functions, α ≥ 1, are meromorphic functions in the unit disk D with α-normal functions (α>1) are characterized by the normality of a family of functions on compact subsets of the finite complex plane . We prove that limit functions of converging sequences of functions {fn (ζ)} are Yosida's functions. We extract a subclass of the α-normal functions such that limit functions of converging sequences of functions are Yosida's functions of the first kind. Note that meromorphic solutions of algebraic differential equations of the first order with coefficients from the Hardy spaces Hp are a-normal functions.
arXiv: Complex Variables, Mar 7, 2007
We discuss the value distribution of quasimeromorphic mappings in R n with given behavior in a ne... more We discuss the value distribution of quasimeromorphic mappings in R n with given behavior in a neighborhood of an essential singularity.
Journal of Mathematical Sciences, 2021
Meromorphic functions with a given growth of a spherical derivative on the complex plane are desc... more Meromorphic functions with a given growth of a spherical derivative on the complex plane are described in terms of the relative location of a-points of functions. The result obtained allows one to construct an example of a meromorphic function in C with a slow growth of Nevanlinna characteristics and arbitrary growth of the spherical derivative. In addition, based on the universality property of the Riemann zeta-function, we estimate the growth of the spherical derivative of ζ(z).
Bulletin of The Australian Mathematical Society, Feb 1, 2009
Functions in the meromorphic Besov, Q p and related classes are characterized in terms of double ... more Functions in the meromorphic Besov, Q p and related classes are characterized in terms of double integrals of certain oscillation quantities involving chordal distances. Some of the results are analogous to the corresponding results in the analytic case.
Complex Variables and Elliptic Equations, Oct 20, 2009
ABSTRACT We study meromorphic functions on the complex plane with given growth of the spherical d... more ABSTRACT We study meromorphic functions on the complex plane with given growth of the spherical derivative. We find a criterion of Lohwater–Pommerenke type for non-weighted Yosida functions, and apply this result to obtain a version of Lappan's 5-point theorem. Further, a necessary condition for all solutions of algebraic differential equation of nth order to be weighted Yosida functions is found.
Computational Methods and Function Theory, Sep 20, 2007
The growth of the Dirichlet integral for functions belonging to Q p but not to the Dirichlet spac... more The growth of the Dirichlet integral for functions belonging to Q p but not to the Dirichlet space is considered. These results are shown to be sharp in a certain sense.
Doklady Mathematics, 2000
Complex Variables and Elliptic Equations, Sep 1, 2009
ABSTRACT Let : [0, 1) → (0, ∞) be an increasing function. A meromorphic function f in the unit di... more ABSTRACT Let : [0, 1) → (0, ∞) be an increasing function. A meromorphic function f in the unit disc is said to be -normal if its spherical derivative f #(z) |f′(z)|/(1 + |f(z)|2) satisfies f #(z) = ((|z|)) as |z| → 1−. This article is devoted to the study of meromorphic -normal functions. In particular, an analogue of the Lohwater–Pommerenke theorem and several equivalent characterizations for -normal functions are established under certain regularity conditions on .
Canadian mathematical bulletin, Mar 1, 2004
The Q p spaces coincide with the Bloch space for p > 1 and are subspaces of BMOA for 0 < p ≤ 1. W... more The Q p spaces coincide with the Bloch space for p > 1 and are subspaces of BMOA for 0 < p ≤ 1. We obtain lower and upper estimates for the essential norm of a composition operator from the Bloch space into Q p , in particular from the Bloch space into BMOA.
Proceedings of the Japan Academy. Series A, Mathematical sciences, 1996
1. Introduction. Let H(D) be the space of analytic functions on the unit disk D. Every holomorphi... more 1. Introduction. Let H(D) be the space of analytic functions on the unit disk D. Every holomorphic self-map o:D-+ D induces a linear composition operator C from H(D) into itself as follows: Cf-f q), whenever f H(D). In this paper we consider composition operators from the Bloch space to the spaces of analytic Besov functions B, 1 < p < oo.
Hokkaido Mathematical Journal, Feb 1, 1997
Compactness of composition operators from the Bloch space B into the analytic Besov spaces B_{p} ... more Compactness of composition operators from the Bloch space B into the analytic Besov spaces B_{p} is characterized by the behavior of the hyperbolic derivative of self-maps of the unit disk D .
Journal of Mathematical Sciences, 2021
Meromorphic functions with a given growth of a spherical derivative on the complex plane are desc... more Meromorphic functions with a given growth of a spherical derivative on the complex plane are described in terms of the relative location of a-points of functions. The result obtained allows one to construct an example of a meromorphic function in C with a slow growth of Nevanlinna characteristics and arbitrary growth of the spherical derivative. In addition, based on the universality property of the Riemann zeta-function, we estimate the growth of the spherical derivative of ζ(z).
Bulletin of The Australian Mathematical Society, Aug 1, 2000
We characterise the compact composition operators from any Mobius invariant Banach space to VMOA,... more We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.
arXiv (Cornell University), Mar 7, 2007
We discuss the value distribution of quasimeromorphic mappings in R n with given behavior in a ne... more We discuss the value distribution of quasimeromorphic mappings in R n with given behavior in a neighborhood of an essential singularity.
Tohoku Mathematical Journal, Dec 1, 2020
We establish pointwise growth estimates for the spherical derivative of solutions of the first or... more We establish pointwise growth estimates for the spherical derivative of solutions of the first order algebraic differential equations. A generalization of this result to higher order equations is also given. We discuss the related question of when for a given class X of meromorphic functions in the unit disc, defined by means of the spherical derivative, and m ∈ N \ {1}, f m ∈ X implies f ∈ X. An affirmative answer to this is given for example in the case of UBC, the α-normal functions with α ≥ 1 and certain (sufficiently large) Dirichlet type classes.
The New York Journal of Mathematics, 2011
Blaschke products are used to construct concrete examples of analytic functions with good integra... more Blaschke products are used to construct concrete examples of analytic functions with good integrability and bad behavior of spherical derivative. These examples are used to show that none of the classes M # p , 0 < p < ∞, is contained in the α-normal class N α when 0 < α < 2. This implies that M # p is in a sense a much larger class than Q # p .
Nihonkai mathematical journal, Dec 1, 1999
We study the cluster sets for analytic functions in the unit disk. Lindel\"of and Meier types the... more We study the cluster sets for analytic functions in the unit disk. Lindel\"of and Meier types theorems are proved for analytic cluster sets.
Journal of Mathematical Sciences, Aug 9, 2019
In this paper, we consider classes of analytical functions that map the unit disk into itself. Fu... more In this paper, we consider classes of analytical functions that map the unit disk into itself. Functions of these classes can be described in terms of hyperbolic derivative and hyperbolic metric. Under an appropriate choice of the corresponding metrics, these classes are metric spaces. Functions of the hyperbolic classes considered generate composition operators from the Bloch space into classical spaces of analytical functions in the unit disk.
Complex Variables, Feb 1, 2001
ABSTRACT α-normal functions, α ≥ 1, are meromorphic functions in the unit disk D with α-normal fu... more ABSTRACT α-normal functions, α ≥ 1, are meromorphic functions in the unit disk D with α-normal functions (α>1) are characterized by the normality of a family of functions on compact subsets of the finite complex plane . We prove that limit functions of converging sequences of functions {fn (ζ)} are Yosida's functions. We extract a subclass of the α-normal functions such that limit functions of converging sequences of functions are Yosida's functions of the first kind. Note that meromorphic solutions of algebraic differential equations of the first order with coefficients from the Hardy spaces Hp are a-normal functions.
arXiv: Complex Variables, Mar 7, 2007
We discuss the value distribution of quasimeromorphic mappings in R n with given behavior in a ne... more We discuss the value distribution of quasimeromorphic mappings in R n with given behavior in a neighborhood of an essential singularity.
Journal of Mathematical Sciences, 2021
Meromorphic functions with a given growth of a spherical derivative on the complex plane are desc... more Meromorphic functions with a given growth of a spherical derivative on the complex plane are described in terms of the relative location of a-points of functions. The result obtained allows one to construct an example of a meromorphic function in C with a slow growth of Nevanlinna characteristics and arbitrary growth of the spherical derivative. In addition, based on the universality property of the Riemann zeta-function, we estimate the growth of the spherical derivative of ζ(z).