Let $G(k)=\int_0^1g(x)e^{kx}dx$, $g\in L^1(0,1)$. The main result of this paper is the following theorem.
{\bf Theorem}. {\it If $\limsup_{k\to +\infty}|G(k)|<\infty$, then $g=0$. There exists $g\not\equiv 0$, $g\in L^1(0,1)$, such that $G(k_j)=0$, $k_j<k_{j+1}$, $\lim_{j\to \infty}k_j=\infty$, $\lim_{k\to \infty}|G(k)|$ does not exist, $\limsup_{k\to +\infty}|G(k)|=\infty$. This $g$ oscillates infinitely often in any interval $[1-δ, 1]$, however small $δ>0$ is.}
The uniqueness problems on transcendental meromorphic or entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results have been obtained. In this paper, we study a transcendental entire function f that shares a non-zero polynomial a with f', together with its linear differential polynomials of the form with rational function coefficients.
In this paper, we continue to investigate the uniqueness problem when an entire function $f$ and its linear differential polynomial $L(f)$ share two distinct complex values CMW (counting multiplicities in the weak sense) jointly. Also, We investigate the same problem when $f$ and its differential monomial $M(f)$ share two distinct complex values CMW. Our results generalize the recent result of Lahiri (Comput. Methods Funct. Theory, https://doi.org/10.1007/s40315-020-00355-4).
In this paper, we prove a normal criteria for family of meromorphic functions. As an application of that result, we establish a uniqueness theorem for entire function concerning a conjecture of R. Bruck. The above uniqueness theorem is an improvement of a problem studied by L. Z. Yang et. al [14]. However, our method differs the method of L. Z. Yang et. al [14]. We mainly use normal family theory and combine it with Nevanlinna theory instead of using only the Nevanlinna theory as in [14].
In this paper, we investigate the sharing values problem that entire function $f(z)$ and its first order difference operator $Δ_ηf(z)$ share two distinct pairs of finite values IM. We prove: Let $f(z)$ be a non-constant entire function of hyper-order less than $1$, let $η$ be a non-zero complex number, and let $a$ be a nonzero finite number. Then there exists no such entire function so that $ f(z)$ and $Δ_ηf(z)$ share $(0,0)$ and $(a,-a)$ IM. Furthermore, using a result in Wang-Chen-Hu \cite{wch}, we obtain some uniqueness results that when $ f(z)$ and $Δ_ηf(z)$ share $a\neq0$ and $-a$ IM.
In the paper, we have shown that every entire solution of the differential difference equation $Δ_η^{m}f-Q_1=(Δ_η^{k}f-Q_{2})e^{P}$ satisfy hyper order of $f=$degree of $P$ and using this result we prove differential difference counter part Brück conjecture. Also we have proved differential difference counter part Brück conjecture for the special case of hyper order of $f<\frac{1}{2}$.
In this paper, we investigate the uniqueness problem of difference polynomials $f^{n}(z)P(f(z))L_c(f)$ and $g^{n}(z)P(g(z))L_c(g)$, where $L_c(f)=f(z+c)+c_0f(z)$, $P(z)$ is a polynomial with constant coefficients of degree $m$ sharing a small function with the notions of weakly weighted sharing and relaxed weighted sharing and obtained the corresponding results, which improve and extend some recent results due to Sahoo and Biswas (Tamkang J. Math., \textbf{49}(2), 85--97 (2018)).
We establish a general uniqueness theorem for subharmonic functions of several variables on a domain. A corollary from this uniqueness theorem for holomorphic functions is formulated in terms of the zero subset of holomorphic functions and restrictions on the growth of functions near the boundary of domain.
In this paper, we extend the uniqueness theorem for meromorphic mappings to the case where the family of hyperplanes depends on the meromorphic mapping and where the meromorphic mappings may be degenerate.
In this paper, we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces (totally geodesic hypersurfaces with respect to a meromorphic connection) in the complex projective space $\mathbb{P}^k$. This is a generalization of Cartan's Second Main Theorem. As a consequence, we establish a uniqueness theorem for holomorphic mappings which intersects $O(k^3)$ many totally geodesic hypersurfaces.
In this article, by introducing a new method in estimating the counting function of the auxiliary function, we prove a new generalization of uniqueness theorems for meromorphic mappings into $¶^n(\C )$ which share few hyperplanes regardless of multiplicities. Our result improves the previous result in this topic. Moreover our proof is simpler than the previous proofs.
In this paper we prove a uniqueness theorem on generalized Kahler-Einstein metrics on Fano manifolds. Our result generalize the one shown by Berndtsson using the convexity properties of Bergman kernels. The same technics as well as that of the regularization of closed positive currents and an estimate due to Demailly-Kollar will be used in our proof.
The purpose of this article is to study the uniqueness problem for meromorphic mappings from $\mathbb{C}^{n}$ into the complex projective space $\mathbb{P}^{N}(\mathbb{C}).$ By making using of the method of dealing with multiple values due to L. Yang and the technique of Dethloff-Quang-Tan respectively, we obtain two general uniqueness theorems which improve and extend some known results of meromorphic mappings sharing hyperplanes in general position.
In this article, we establish some new second main theorems for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ and moving hypersurfaces with truncated counting functions. A uniqueness theorem for these mappings sharing few moving hypersurfaces without counting multiplicity is also given. This result is an improvement of the recent result of Dethloff - Tan [3]. Moreover the meromorphic mappings in our result may be algebraically degenerate. The last purpose of this article is to study uniqueness problem in the case where the meromorphic mappings agree on small identical set…
For a scale of spaces $X$ of functions analytic in the unit disc, including the Korenblum space, and for some natural families $\mathcal E$ of uniqueness subsets for $X$, we describe minorants for $(X,\mathcal E)$, that is non-decreasing functions $M:(0,1)\to(0,\infty)$ such that $f\in X$, $E\in\mathcal E$, and $\log|f(z)|\le -M(|z|)$ on $E$ imply $f=0$. We give an application of this result to approximation by simple fractions with restrictions on the coefficients.
Uniqueness theorems are considered for various types of almost periodic objects: functions, measures, distributions, multisets, holomorphic and meromorphic functions.
We obtain uniqueness theorems for harmonic and subharmonic functions of a new type. They lead to new analytic extension criteria and new conditions for stability of operator semigroups in Banach spaces with Fourier type.
This work establishes a strong uniqueness property for a class of planar locally integrable vector fields. A result on pointwise convergence to the boundary value is also proved for bounded solutions.
The purpose of this article is to show uniqueness theorems for meromorphic mappings of C^m to CP^n with few hyperplanes H_j, j=1,...,q. It is well known that uniqueness theorems hold for q \geq 3n+2. In this paper we show that for every nonnegative integer c there exists a positive integer N(c), depending only on c in an explicit way, such that uniqueness theorems hold if q\geq (3n+2 -c) and n\geq N(c). Furthermore, we also show that the coefficient of n in the formula of q can be replaced by a number which is strictly smaller than 3 for all n>>0. At the same time, a big number of recent uniqu…
We recall a uniqueness theorem of E. B. Vul pertaining to a version of the cosine transform originating in spectral theory. Then we point out an application to the Bernstein approximation problem with non-symmetric weights: a theorem of Volberg is proved by elementary means.
In this paper, we establish a uniqueness theorem for algebraically nondegenerate meromorphic maps of C^m into C P^n and slowly moving hypersurfaces Q_j in C P^n, j=1,...,q in (weakly) general position, where q depends effectively on n and on the degrees d_j of the hypersurfaces Q_j.
For $ϕ$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $$ \int_X e^{-ϕ}. $$ We prove that the logarithm of the volume is concave along continuous geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on $X$. As consequences we get a simplified proof of the Bando-Mabuchi uniqueness theorem for Kähler - Einstein metrics and a generalization of this theorem to 'twisted' Kähler-Einstein metrics.
Let $μ$ be a positive finite Borel measure on the unit circle. The associated Dirichlet space $\mathcal{D}(μ)$ consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of $μ$. We give a sufficient condition on a Borel subset $E$ of the unit circle which ensures that $E$ is a uniqueness set for $\mathcal{D}(μ)$. {We also give somes examples of positive Borel measures $μ$ and uniqueness sets for $\mathcal{D}(μ)$.}
In this paper, we give some results on the number of meromorphic mappings of C^m into P^n under a condition on the inverse images of hyperplanes in P^n. At the same time, we give an answer for an open question by H.Fujimoto.
We rereview the classical Cauchy-Kovalevskaya theorem and the related uniqueness theorem of Holmgren, in the simple setting of powers of the Laplacian and a smooth curve segment in the plane. As a local problem, the Cauchy-Kovalevskaya and Holmgren theorems supply a complete answer to the existence and uniqueness issues. Here, we consider a global uniqueness problem of Holmgren's type. Perhaps surprisingly, we obtain a connection with the theory of quadrature identities, which demonstrates that rather subtle algebraic properties of the curve come into play. For instance, if $Ω$ is the interior…
For $ϕ$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $$ \int_X e^{-ϕ}. $$ We prove that the logarithm of the volume is concave along bounded geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on $X$. As a consequence we get a simplified proof of the Bando-Mabuchi uniqueness theorem for Kähler - Einstein metrics. A generalization of this theorem to 'twisted' Kähler-Einstein metrics and some classes of manifolds that satisfy weake…
We establish a Fatou-type Theorem for $J$-holomorphic mappings that are bounded in an appropriate sense and we prove the Blaschke condition for their zero sets. We also prove a Privalov-type uniqueness Theorem for pairs of $J$-holomorphic mappings.
We prove that there exists a nonzero holderian real-to-real function vanishing together with its M. Rietz potential in all points of some set of positive length. This result improves the one of D. Beliaev and V. Havin. We also extend the results to multidimensional M. Rietz potentials.
Two measurable sets $S, Λ\subseteq \mathcal{R}^d$ form a Heisenberg uniqueness pair, if every bounded measure $μ$ with support in S whose Fourier transform vanishes on Λ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathcal{R}^d$. As a corollary we obtain a new, surprising version of the classical Cramér-Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of…
