Recent zbMATH articles in MSC 30D15https://www.zbmath.org/atom/cc/30D152022-05-16T20:40:13.078697ZWerkzeugA bilogarithmic criterion for the existence of a regular minorant that does not satisfy the bang conditionhttps://www.zbmath.org/1483.300072022-05-16T20:40:13.078697Z"Gaisin, R. A."https://www.zbmath.org/authors/?q=ai:gaisin.rashit-akhtyarovichSummary: Problems of constructing regular majorants for sequences \(\mu=\{\mu_n\}_{n=0}^{\infty}\) of numbers \(\mu_n\ge0\) that are the Taylor coefficients of integer transcendental functions of minimal exponential type are investigated. A new criterion for the existence of regular minorants of associated sequences of the extended half-line \((0,+\infty]\) in terms of the Levinson bilogarithmic condition \(M=\{\mu_n^{-1}\}_{n=0}^{\infty}\) is obtained. The result provides a necessary and sufficient condition for the nontriviality of the important subclass defined by J. A. Siddiqi. The proofs of the main statements are based on properties of the Legendre transform.Generalizations and sharpenings of certain Bernstein and Turán types of inequalities for the polar derivative of a polynomialhttps://www.zbmath.org/1483.300252022-05-16T20:40:13.078697Z"Singh, Thangjam Birkramjit"https://www.zbmath.org/authors/?q=ai:singh.thangjam-birkramjit"Chanam, Barchand"https://www.zbmath.org/authors/?q=ai:chanam.barchandSummary: Let \(p(z)\) be a polynomial of degree \(n\). The polar derivative of \(p(z)\) with respect to a complex number \(\alpha\) is defined by
\[
D_\alpha p(z)=np(z)+(\alpha-z)p'(z).
\]
If \(p(z)\) has all its zeros in \(|z|\leq k\), \(k\geq 1\), then for \(|\alpha|\geq k\), \textit{A. Aziz} and \textit{N. A. Rather} [Math. Inequal. Appl. 1, No. 2, 231--238 (1998; Zbl 0911.30002)] proved
\[
\max\limits_{|z|=1}|D_\alpha p(z)|\geq n\left(\frac{|\alpha|-k}{1+k^n}\right)\max\limits_{|z|=1}|p(z)|.
\]
In this paper, we first improve as well as generalize the above inequality. Besides, we are able to prove an improvement of a result due to \textit{N. K. Govil} and \textit{G. N. McTume} [Acta Math. Hung. 104, No. 1--2, 115--126 (2004; Zbl 1060.30004)] and also prove an inequality for a subclass of polynomials having all its zeros in \(|z|\geq k\), \(k\leq 1\).The use of the isometry of function spaces with different numbers of variables in the theory of approximation of functionshttps://www.zbmath.org/1483.300582022-05-16T20:40:13.078697Z"Bushev, D. M."https://www.zbmath.org/authors/?q=ai:bushev.d-m"Abdullayev, F. G."https://www.zbmath.org/authors/?q=ai:abdullayev.fahreddin-g"Kal'chuk, I. V."https://www.zbmath.org/authors/?q=ai:kalchuk.inna-v"Imashkyzy, M."https://www.zbmath.org/authors/?q=ai:imashkyzy.meerimSummary: In the work, we found integral representations for function spaces that are isometric to spaces of entire functions of exponential type, which are necessary for giving examples of equality of approximation characteristics in function spaces isometric to spaces of non-periodic functions. Sufficient conditions are obtained for the nonnegativity of the delta-like kernels used to construct isometric function spaces with various numbers of variables.Rate of growth of distributionally chaotic functionshttps://www.zbmath.org/1483.300592022-05-16T20:40:13.078697Z"Gilmore, Clifford"https://www.zbmath.org/authors/?q=ai:gilmore.clifford"Martínez-Giménez, Félix"https://www.zbmath.org/authors/?q=ai:martinez-gimenez.felix"Peris, Alfred"https://www.zbmath.org/authors/?q=ai:peris.alfredoSummary: We investigate the permissible growth rates of functions that are distributionally chaotic with respect to differentiation operators. We improve on the known growth estimates for \(D\)-distributionally chaotic entire functions, where growth is in terms of average \(L^p\)-norms on spheres of radius \(r>0\) as \(r\rightarrow\infty\), for \(1\leq p\leq\infty\). We compute growth estimates of \(\partial/\partial x_k\)-distributionally chaotic harmonic functions in terms of the average \(L^2\)-norm on spheres of radius \(r>0\) as \(r\rightarrow\infty\). We also calculate sup-norm growth estimates of distributionally chaotic harmonic functions in the case of the partial differentiation operators \(D^\alpha\).Global boundedness of functions of finite order that are bounded outside small setshttps://www.zbmath.org/1483.300602022-05-16T20:40:13.078697Z"Khabibullin, Bulat N."https://www.zbmath.org/authors/?q=ai:khabibullin.b-nOn the number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficientshttps://www.zbmath.org/1483.300612022-05-16T20:40:13.078697Z"Nguyen, Thu Hien"https://www.zbmath.org/authors/?q=ai:nguyen.thu-hien-thi"Vishnyakova, Anna"https://www.zbmath.org/authors/?q=ai:vishnyakova.anna-mSummary: For an entire function \(f(z)=\sum^\infty_{k=0}a_kz^k\), \(a_k>0\), we define the sequence of the second quotients of Taylor coefficients \(Q:=\bigg(\frac{a^2_k}{a_{k-1}a_{k+1}}\bigg)^\infty_{k=1}\). We find new necessary conditions for a function with a non-decreasing sequence \(Q\) to belong to the Laguerre-Pólya class of type I. We also estimate the possible number of non-real zeros for a function with a non-decreasing sequence \(Q\).Invariant spaces of entire functionshttps://www.zbmath.org/1483.460212022-05-16T20:40:13.078697Z"Krivosheev, A. S."https://www.zbmath.org/authors/?q=ai:krivosheev.aleksandr-sergeevich"Krivosheeva, O. A."https://www.zbmath.org/authors/?q=ai:krivosheeva.o-aLet \(D\subset\mathbb{C}\) be a convex domain and let \(H(D)\) be the space of holomorphic functions on \(D\) endowed with the compact open topology. The paper under review deals with the following problem: Let \(W\) be an invariant subspace of the differentiation operator on \(H(D)\). Which conditions ensure that all functions of \(W\) can be extended to entire functions? This problem naturally arises from the problem of expanding convergence domains of exponential series and their special cases, power series and Dirichlet series. \(W\) is assumed to satisfy \textit{spectral synthesis}, i.e., the closure of the span of the eigenvectors of the differentiation operator in \(H(D)\) is the whole \(W\). The following subset of the unit circle \(\mathbb{T}\) is defined,
\[
J(D)= \Bigl\{\omega\in \mathbb T: \ \sup_{z\in D}\text{Re}\,z\omega=+\infty \Bigr\}.
\]
Let \(\Delta:=\{\lambda_k: k\in\mathbb{N}\}\) be the sequence of eigenvalues of the differentiation operator acting on \(W\). Let \(\Xi(\Delta):=\{\overline{\lambda}/|\lambda|: \lambda\in \Delta\} \). The main theorem asserts that the continuation problem has a positive solution when \(\Xi(\Delta)\subset J(D)\). This result was known only under the assumption that \(J(D)\) is open in \(\mathbb{T}\).
Reviewer: Enrique Jordá (Alicante)