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.

It is known that if $f$ is holomorphic in the open unit disc $𝔻$ of the complex plane and if, for some $c>0$, $\left|f\left(z\right)\right|\le {1/(1-|z|}^2{)}^c$, $z\in 𝔻$, then $|f^{\text{'}}{\left(z\right)|\le 2(c+1)/(1-\left|z\right|}^2{)}^{c+1}$. We consider a meromorphic analogue of this result. Furthermore, we introduce and study the class of meromorphic Bloch-type functions that possess a nonzero simple pole in $𝔻$. In particular, we obtain bounds for the modulus of the Taylor coefficients of functions in this class.

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...

The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation $$f^n\left(z\right)+P_d(z,f)=p_1\left(z\right){\mathrm{e}}^{{\alpha }_1\left(z\right)}+p_2\left(z\right){\mathrm{e}}^{{\alpha }_2\left(z\right)},$$ where $P_d(z,f)$ is a difference-differential polynomial in $f\left(z\right)$ of degree $d\le n-1$ with small functions of $f\left(z\right)$ as its coefficients, $p_1$, $p_2$ are nonzero rational functions and ${\alpha }_1$, ${\alpha }_2$ are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation.

Firstly we study the growth of meromorphic solutions of linear difference equation of the form$$A_k\left(z\right)f(z+c_k)+\cdots +A_1\left(z\right)f(z+c_1)+A_0\left(z\right)f\left(z\right)=F\left(z\right),$$ where $A_k\left(z\right),...,A_0\left(z\right)$ and $F\left(z\right)$ are meromorphic functions of finite logarithmic order, $c_i$ $(i=1,...,k,k\in \mathbb{N})$ are distinct nonzero complex constants. Secondly, we deal with the growth of solutions of differential-difference equation of the form $$\sum _{i=0}^n\sum _{j=0}^m{A}_{ij}\left(z\right)f^{\left(j\right)}(z+c_i)=F\left(z\right),$$ where $A_{ij}\left(z\right)$ $(i=0,1,...,n,j=0,1,...,m,n,m\in \mathbb{N})$ and $F\left(z\right)$ are meromorphic functions of finite logarithmic order, $c_i$ $(i=0,...,n)$ are distinct complex constants. We extend some previous results...

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...

With the idea of normal family we study the uniqueness of meromorphic functions $f$ and $g$ when $f^n{\left(f^{\left(k\right)}\right)}^m-p$ and $g^n{\left(g^{\left(k\right)}\right)}^m-p$ share two values, where $p$ is any nonzero polynomial. The result of this paper significantly improves and generalizes the result due to A. Banerjee and S. Majumder (2018).

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.

This paper deals with the finiteness problem of meromorphic funtions on an annulus sharing four values regardless of multiplicity. We prove that if three admissible meromorphic functions $f_1$, $f_2$, $f_3$ on an annulus $𝔸\left(R_0\right)$ share four distinct values regardless of multiplicity and have the of positive counting function, then $f_1=f_2$ or $f_2=f_3$ or $f_3=f_1$. This result deduces that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicity truncated to level $2$ and sharing...

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...........................................................