Related papers: Macdonald's Theorem for Analytic Functions

In this note we give some sufficient conditions for an analytic function $f(z)$ normalized by $f'(0)=1$ to belong to certain subfamilies of the class of Bazilevic functions. In earlier works, the closure property of many classes of…

Using an annular version of the F. and M. Riesz theorem, we prove a generalization of the Rudin-Carleson theorem for finitely connected bounded domains. That is, for a continuous function on a closed set in the boundary of measure zero…

We obtain an estimate for uniform approximation rate of bounded analytic in the unit disk functions by logarithmic derivatives of $C$-polynomials, i.e., polynomials, all of whose zeros lie on the unit circle $C:|z|=1$.

We give the following version of Fatou's theorem for distributions that are boundary values of analytic functions. We prove that if $f\in\mathcal{D}^{\prime}(a,b) $ is the distributional limit of the analytic function $F$ defined in a…

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $f_1(z),\ldots, f_m(z) \in K[[z]]$ such that, for every $1 \leq i \leq m$, $f_i(z)$ is a solution of a differential operator $\mathcal{L}_i \in E_p[d/dz]$, where $E_p$ is the field of…

For analytic functions f(z) in the closed unit disk \bar{U}, two boundary points z_1 and z_2 such that \alpha = (f'(z_1)+f'(z_2))/2 in f'(U) are considered. The object of the present paper is to discuss some interesting conditions for f(z)…

In the seventies', Zehnder found a Nash-Moser type implicit function theorem in the analytic set-up. This theorem has found many applications in dynamical systems although its applications require, as a general rule, some efforts. We…

For a fixed analytic function $g$ on the unit disc $\mathbb{D}$, we consider the analytic paraproducts induced by $g$, which are defined by $T_gf(z)= \int_0^z f(\zeta)g'(\zeta)\,d\zeta$, $S_gf(z)= \int_0^z f'(\zeta)g(\zeta)\,d\zeta$, and…

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…

If all the zeros of $n$th degree polynomials $f(z)$ and $g(z) = \sum_{k=0}^{n}\lambda_k\binom{n}{k}z^k$ respectively lie in the cricular regions $|z|\leq r$ and $|z| \leq s|z-\sigma|$, $s>0$, then it was proved by Marden \cite[p. 86]{mm}…

Let ${\mathcal U}(\lambda)$ denote the family of analytic functions $f(z)$, $f(0)=0=f'(0)-1$, in the unit disk $\ID$, which satisfy the condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<\lambda $ for some $0<\lambda \leq 1$. The…

We prove that there is a continuous non-negative function $g$ on the unit sphere in $\cd$, $d \geq 2$, whose logarithm is integrable with respect to Lebesgue measure, and which vanishes at only one point, but such that no non-zero bounded…

The main objects of study in this paper are those functionals that are analytic in the sense that they annihilate the non-commutative disc algebra. In the classical univariate case, a theorem of F. and M. Riesz implies that such functionals…

Let A_n be the class of functions f(z) which are analytic in the open unit disk U} with f(0)=0, f'(0)=1, f"(0)=f"'(0)=...=f^{(n)}=0 and f^{(n+1)}\neq0. Applying the results due to S. S. Miller (J. Math. Anal. Appl. 65(1978), 289-305), some…

We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $\alpha$, with $0<\alpha<1$, in the vicinity of an exceptional boundary point where all such functions…

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free…

Under very general conditions it is shown that if $A$ is a uniform algebra generated by real-analytic functions, then either $A$ consists of all continuous functions or else there exists a disc on which every function in $A$ is holomorphic.…

Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to…

In classical function theory, a function is holomorphic if and only if it is complex analytic. For higher dimensional spaces it is natural to work in the context of Clifford algebras. The structures of these algebras depend on the parity of…

In this paper we define a new concept of quasi-convolution for analytic functions normalized by $f(0)=0$ and $f^\prime(0)=1$ in the unit disk $E=\{z\in \mathbb{C}\colon |z|<1\}$. We apply this new approach to study the closure properties of…