We prove the existence of sequences ${\varrho ₙ}_{n=1}^{\infty }$, ϱₙ → 0⁺, and ${zₙ}_{n=1}^{\infty }$, |zₙ| = 1/2, such that for every α ∈ ℝ and for every meromorphic function G(z) on ℂ, there exists a meromorphic function $F\left(z\right)=F_{G,\alpha }\left(z\right)$ on ℂ such that $\varrho {ₙ}^{\alpha }F(nzₙ+n\varrho ₙ\zeta )$ converges to G(ζ) uniformly on compact subsets of ℂ in the spherical metric. As a result, we construct a family of functions meromorphic on the unit disk that is $Q_m$-normal for no m ≥ 1 and on which an extension of Zalcman’s Lemma holds.

Let $X$ and $Y$ be compact Kähler manifolds, and let $f:X\to Y$ be a dominant meromorphic map. Based upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator $f^{♯}$ for currents of bidegrees $(p,p)$ of finite order on $Y$ (and thus forcurrent, since $Y$ is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony,...

The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and...

Let $k$ be a nonnegative integer or infinity. For $a\in ℂ\cup \left\{\infty \right\}$ we denote by $E_k(a;f)$ the set of all $a$-points of $f$ where an $a$-point of multiplicity $m$ is counted $m$ times if $m\le k$ and $k+1$ times if $m>k$. If $E_k(a;f)=E_k(a;g)$ then we say that $f$ and $g$ share the value $a$ with weight $k$. Using this idea of sharing values we study the uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a nonzero polynomial with finite weight. The results of the paper improve and generalize the related results due to...

Let $A$ be the class of analytic functions in the unit disc $U$ of the complex plane $ℂ$ with the normalization $f\left(0\right)=f^{{}^{\text{'}}}\left(0\right)-1=0$. We introduce a subclass $S_M^{*}(\alpha ,b)$ of $A$, which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class $S_M^{*}(n,\alpha ,b)$ ($n\ge 0$) related to $S_M^{*}(\alpha ,b)$ is also considered under the same conditions. Among other things, we find convolution conditions for a function $f\in A$ to belong to the class $S_M^{*}(\alpha ,b)$. Several properties of the class $S_M^{*}(n,\alpha ,b)$ are investigated. ...

We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $\le 1/d$ of the $L$-functions of any degree $d\ge 1$ in the extended Selberg class. In particular, this solves the resonance problem in all such cases.

Let f be a transcendental meromorphic function and $g\left(z\right)=f(z+c₁)+\cdots +f(z+c_k)-kf\left(z\right)$ and $g_k\left(z\right)=f(z+c₁)\cdots f(z+c_k)-f^k\left(z\right)$. A number of results are obtained concerning the exponents of convergence of the zeros of g(z), $g_k\left(z\right)$, g(z)/f(z), and $g_k\left(z\right)/f^k\left(z\right)$.

Let Mₚ denote the class of functions f of the form $f\left(z\right)=1/z^p+{\sum }_{k=0}^{\infty }aₖz^k$, p a positive integer, in the unit disk E = |z| < 1, f being regular in 0 < |z| < 1. Let $L_{n,p}\left(\alpha \right)=f:f\in Mₚ,Re-(z^{p+1}/p){\left(Dⁿf\right)}^{\text{'}}>\alpha$, α < 1, where $Dⁿf={\left(z^{n+p}f\left(z\right)\right)}^{\left(n\right)}/(z^{p}n!)$. Results on $L_{n,p}\left(\alpha \right)$ are derived by proving more general results on differential subordination. These results reduce, by putting p =1, to the recent results of Al-Amiri and Mocanu.

In the present work, we introduce the subclass ${𝒯}_{\gamma ,\alpha }^k\left(\varphi \right)$, of starlike functions with respect to $k$-symmetric points of complex order $\gamma$ ($\gamma \ne 0$) in the open unit disc $$. Some interesting subordination criteria, inclusion relations and the integral representation for functions belonging to this class are provided. The results obtained generalize some known results, and some other new results are obtained.

Let ${𝒫}_n$ denote the class of analytic functions $p\left(z\right)$ of the form $p\left(z\right)=1+c_n{z}^n+c_{n+1}{z}^{n+1}+\cdots$ in the open unit disc $𝕌$. Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for $p\left(z\right)$ concerned with Caratheodory functions are discussed. Further, some corollaries of the results concerned with the result due to M. Obradovic and S. Owa (Math. Nachr. 140 (1989), 97-102) are shown.

For γ ∈ ℂ such that |γ| < π/2 and 0 ≤ β < 1, let ${}_{\gamma ,\beta }$ denote the class of all analytic functions P in the unit disk with P(0) = 1 and $Re\left(e^{i\gamma }P\left(z\right)\right)>\beta cos\gamma$ in . For any fixed z₀ ∈ and λ ∈ ̅, we shall determine the region of variability $V_{}(z₀,\lambda )$ for ${\int }_0^{z₀}P\left(\zeta \right)d\zeta$ when P ranges over the class $\left(\lambda \right)=P{\in }_{\gamma ,\beta }:P^{\text{'}}\left(0\right)=2(1-\beta )\lambda e^{-i\gamma }cos\gamma .$As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.

We consider functions of the type $F\left(z\right)=z{\prod }_{j=1}^n{[f_j\left(z\right)/z]}^{a_j}$, where $a_j$ are real numbers and $f_j$ are ${\beta }_j$-strongly close-to-starlike functions of order ${\alpha }_j$. We look for conditions on the center and radius of the disk (a,r) = z:|z-a| < r, |a| < r ≤ 1 - |a|, ensuring that F((a,r)) is a domain starlike with respect to the origin.

In this paper, we obtain new sufficient conditions for the operators $F_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ and $G_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ to be univalent in the open unit disc $𝒰$, where the functions $f_1,f_2,...,f_n$ belong to the classes $S^{*}(a,b)$ and $𝒦(a,b)$. The order of convexity for the operators $F_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ and $G_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ is also determined. Furthermore, and for $\beta =1$, we obtain sufficient conditions for the operators $F_n\left(z\right)$ and $G_n\left(z\right)$ to be in the class $𝒦(a,b)$. Several corollaries and consequences of the main results are also considered.

The purpose of this paper is to study some properties related to convexity order and coefficients estimation for a general integral operator. We find the convexity order for this operator, using the analytic functions from the class of starlike functions of order $\alpha$ and from the class $\mathrm{𝒞𝒱\mathscr{H}}\left(\beta \right)$ and also we estimate the first two coefficients for functions obtained by this operator applied on the class $\mathrm{𝒞𝒱\mathscr{H}}\left(\beta \right)$.

We consider the class ${\mathscr{H}}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $\left|z\right|<1$ and normalized so that $h\left(0\right)=0=h^{\text{'}}\left(0\right)-1$ and $g\left(0\right)=0=g^{\text{'}}\left(0\right)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes ${𝒫}_H^0\left(\alpha \right)$ and ${𝒢}_H^0\left(\beta \right)$ of functions from ${\mathscr{H}}_0$ and show that if $f\in {𝒫}_H^0\left(\alpha \right)$ and $F\in {𝒢}_H^0\left(\beta \right)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $\alpha$ and $\beta$ are satisfied. In the second part we study the harmonic sections...

Let ${\alpha }_i,{\beta }_i>0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t•x=(t^{\alpha ₁}x₁,...,t^{\alpha ₙ}xₙ)$, $t\circ x=(t^{\beta ₁}x₁,...,t^{\beta ₙ}xₙ)$ and $\left|\right|x\left|\right|={\sum }_{i=1}^n{\left|x_i\right|}^{1/{\alpha }_i}$. Let φ₁,...,φₙ be real functions in $C^{\infty }(ℝⁿ-0)$ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on ${ℝ}^{2n}$ given by $\mu \left(E\right)={\int }_{ℝⁿ}{\chi }_E{(x,\phi \left(x\right))\left|\right|x\left|\right|}^{\gamma -\alpha }dx$, where $\alpha ={\sum }_{i=1}^n{\alpha }_i$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_{\mu }f=\mu \ast f$ and let $\left|\right|T_{\mu }{\left|\right|}_{p,q}$ be the operator norm of $T_{\mu }$ from $L^p\left({ℝ}^{2n}\right)$ into $L^q\left({ℝ}^{2n}\right)$, where the $L^p$ spaces are taken with respect to the Lebesgue measure. The type set $E_{\mu }$ is defined by $E_{\mu }=(1/p,1/q):\left|\right|T_{\mu }{\left|\right|}_{p,q}<\infty ,1\le p,q\le \infty$. In the case ${\alpha }_i\ne {\beta }_k$ for 1 ≤ i,k ≤ n we characterize the...

CONTENTS0. Introduction.......................................................................... 51. Preliminaries............................................................................... 72. Basic facts to be used in the sequel....................................... 113. Predicates OD(.,.) and CL(.,.).................................................... 174. Predicate Sels............................................................................. 185. Strong ${\sum }_n^1$-collection...........................................................

Let $\mathrm{T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta :=\{z\in ℂ:|z|<1\}$, normalized by $f\left(0\right)=f^{\text{'}}\left(0\right)-1=0$ and such that Im $z$ Im $f\left(z\right)$ $\ge 0$ for $z\in \Delta$. Moreover, let us denote: ${\mathrm{T}}^{\left(2\right)}:=\{f\in \mathrm{T}:f\left(z\right)=-f(-z)\text{for}z\in \Delta \}$ and ${\mathrm{T}}^{M,g}:=\{f\in \mathrm{T}:f\prec Mg\text{in}\Delta \}$, where $M>1$, $g\in \mathrm{T}\cap \mathrm{S}$ and $\mathrm{S}$ consists of all analytic functions, normalized and univalent in $\Delta$.We investigate  classes in which the subordination is replaced with the majorization and the function $g$ is typically real but does not necessarily univalent, i.e. classes $\{f\in \mathrm{T}:f\ll Mg\text{in}\Delta \}$, where $M>1$, $g\in \mathrm{T}$, which we denote...

A convolution operator, bounded on $L^q({ℝ}^n)$, is bounded on $L^p({ℝ}^n)$, with the same operator norm, if $p$ and $q$ are conjugate exponents. It is well known that this fact is false if we replace ${ℝ}^n$ with a general non-commutative locally compact group $G$. In this paper we give a simple construction of a convolution operator on a suitable compact group $G$, wich is bounded on $L^q(G)$ for every $q\in [2,\mathrm{\infty })$ and is unbounded on $L^p(G)$ if $p\in [1,2)$.

Sia $K\subset \subset {ℝ}^n$ un compatto, $f\ge 0$ una funzione analitica all'intorno di $K$, ed $m$ la massima molteplicità in $K$ degli zeri di $f$; si prova che la potenza $f^{\lambda }$ ($\lambda \in ℂ$, $Re\lambda >\mathrm{—}\frac{1}{m}$) è integrabile in $K$. L'estensione meromorfa dell'applicazione $\lambda \to f^{\lambda }$ da $Re\lambda >0$ a tutto $ℂ$ (con valori in ${𝒟}^{\prime }(K)$ anziché in $L^1(K)$) era già stata provata in [1] e [2].

The class of $H^{\left(N\right)}$-sets forms an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of this class which is reflected by the family of measures (called polar) annihilating all sets from the class. The main aim of this paper is to answer in the negative a question stated by Lyons, whether the polars of the classes of $H^{\left(N\right)}$-sets are the same for all N ∈ ℕ. To prove our result we also present a new description of $H^{\left(N\right)}$-sets.

We investigate the question whether a system ${\left(E_i\right)}_{i\in I}$ of homogeneous linear equations over $\mathbb{Z}$ is non-trivially solvable in $\mathbb{Z}$ provided that each subsystem ${\left(E_j\right)}_{j\in J}$ with $\left|J\right|\le c$ is non-trivially solvable in $\mathbb{Z}$ where $c$ is a fixed cardinal number such that $c<\left|I\right|$. Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., $I$ being finite). (b) The answer is ‘No’ in the denumerable case (i.e., $\left|I\right|={\aleph }_0$ and $c$ a natural number). (c) The answer in case that $I$ is uncountable and $c\le {\aleph }_0$ is ‘No...

This note presents the study of the conjugacy classes of elements of some given finite order $n$ in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if $n$ is even, $n=3$ or $n=5$, and that it is equal to $3$ (respectively $9$) if $n=9$ (respectively if $n=15$) and to $1$ for all remaining odd orders. Some precise representative elements of the classes are given.

Let $\mu \in {\left({ℝ}^d\right)}^{\text{'}}$ be an analytic functional and let $T_{\mu }$ be the corresponding convolution operator on Sato’s space $\left({ℝ}^d\right)$ of hyperfunctions. We show that $T_{\mu }$ is surjective iff $T_{\mu }$ admits an elementary solution in $\left({ℝ}^d\right)$ iff the Fourier transform μ̂ satisfies Kawai’s slowly decreasing condition (S). We also show that there are $0\ne \mu \in {\left({ℝ}^d\right)}^{\text{'}}$ such that $T_{\mu }$ is not surjective on $\left({ℝ}^d\right)$.