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Register a new userHypertranscendency of Perturbations of Hypertranscendental Functions
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Inspired by the work of Bank on the hypertranscendence of $Gamma e^h$ where $Gamma$ is the Euler gamma function and $h$ is an entire function, we investigate when a meromorphic function $fe^g$ cannot satisfy any algebraic differential equation over certain field of meromorphic functions, where $f$ and $g$ are meromorphic and entire on the complex plane, respectively. Our results (Theorem 1 and 2) give partial solutions to Banks Conjecture (1977) on the hypertranscendence of $Gamma e^h$. We also give some sufficient conditions for hypertranscendence of meromorphic function of the form $f+g$, $fcdot g$ and $fcirc g$ in Theorem 3 and 4.
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V. Nestoridis conjectured that if $Omega$ is a simply connected subset of $mathbb{C}$ that does not contain $0$ and $S(Omega)$ is the set of all functions $fin mathcal{H}(Omega)$ with the property that the set $left{T_N(f)(z)coloneqqsum_{n=0}^Ndfrac{f^{(n)}(z)}{n!} (-z)^n : N = 0,1,2,dots right}$ is dense in $mathcal{H}(Omega)$, then $S(Omega)$ is a dense $G_delta$ set in $mathcal{H}(Omega)$. We answer the conjecture in the affirmative in the special case where $Omega$ is an open disc $D(z_0,r)$ that does not contain $0$.
Given two compact sets, $E$ and $F$, on the unit circle, we study the class of subharmonic functions on the unit disk which can grow at the direction of $E$ and $F$ (sets of singularities) at different rate. The main result concerns the Blaschke-type condition for the Riesz measure of such functions. The optimal character of such condition is demonstrated.
If $f$ is an entire function and $a$ is a complex number, $a$ is said to be an asymptotic value of $f$ if there exists a path $gamma$ from $0$ to infinity such that $f(z) - a$ tends to $0$ as $z$ tends to infinity along $gamma$. The Denjoy--Carleman--Ahlfors Theorem asserts that if $f$ has $n$ distinct asymptotic values, then the rate of growth of $f$ is at least order $n/2$, mean type. A long-standing problem asks whether this conclusion holds for entire functions having $n$ distinct asymptotic (entire) functions, each of growth at most order $1/2$, minimal type. In this paper conditions on the function $f$ and associated asymptotic paths are obtained that are sufficient to guarantee that $f$ satisfies the conclusion of the Denjoy--Carleman--Ahlfors Theorem. In addition, for each positive integer $n$, an example is given of an entire function of order $n$ having $n$ distinct, prescribed asymptotic functions, each of order less than $1/2$.
A description of the Bloch functions that can be approximated in the Bloch norm by functions in the Hardy space $H^p$ of the unit ball of $Cn$ for $0<p<infty$ is given. When $0<pleq1$, the result is new even in the case of the unit disk.
We establish a weighted inequality for the Bergman projection with matrix weights for a class of pseudoconvex domains. We extend a result of Aleman-Constantin and obtain the following estimate for the weighted norm of $P$: [|P|_{L^2(Omega,W)}leq C(mathcal B_2(W))^{{2}}.] Here $mathcal B_2(W)$ is the Bekolle-Bonami constant for the matrix weight $W$ and $C$ is a constant that is independent of the weight $W$ but depends upon the dimension and the domain.
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