scholarly journals The space of totally bounded analytic functions

1987 ◽  
Vol 30 (2) ◽  
pp. 229-246
Author(s):  
Alan L. Horwitz ◽  
Lee A. Rubel
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Bounded Analytic Functions ◽  
Sharp Bounds ◽  
Totally Bounded ◽  
Bounded Set ◽  
Inverse Interpolation ◽  
Derivatives Of ◽  
Uniformly Bounded

This paper is a continuation of our project on “inverse interpolation”, begun in [6]. In brief, the task of inverse interpolation is to deduce some property of a function f from some given property of the set L of its Lagrange interpolants. In the present work, the property of L is that it be a uniformly bounded set of functions when restricted to the domain of f. In particular (see Section 3), when the domain is a disc, we deduce sharp bounds on the successive derivatives of f. As a result, f must extend to be an analytic function (of restricted growth) in the concentric disc of thrice the original radius.

scholarly journals Sharp estimates of generalized zalcman functional of early coeffcients for Ma-Minda type functions

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6267-6280 ◽  
Author(s):  
Nak Cho ◽  
Oh Kwon ◽  
Adam Lecko ◽  
Young Sim
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sharp Bounds ◽  
Sharp Estimates

Let ? be an analytic function in the unit disk D := {z ? C : |z| < 1} which has the form ?(z) = 1 + p1z + p2z2 + p3z3 + ... with p1 > 0, p2, p3 ? R. For given such ?, let S*(?), K(?) and R(?) denote the classes of standardly normalized analytic functions f in D which satisfy zf'(z)/f(z) < ?(z), 1 + zf''(z)/f'(z) < ?(z) f'(z)< ?(z), z ? D, respectively, where < means the usual subordination. In this paper, we find the sharp bounds of |a2a3-a4|, where an := f(n)(0)/n!; n ? N, over classes S*(?), K(?) and R(?).

scholarly journals Bounded analytic functions and the little Bloch space

1990 ◽  
Vol 13 (1) ◽  
pp. 193-198 ◽  
Author(s):  
Rohan Attele
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Bloch Space ◽  
Bounded Analytic Functions ◽  
Radial Limits ◽  
Bounded Analytic Function ◽  
Little Bloch Space

The radial limits of the weighted derivative of an bounded analytic function is considered.

scholarly journals A discrete analogue of the Paley-Wiener theorem for bounded analytic functions in a half plane

1977 ◽  
Vol 23 (3) ◽  
pp. 376-378
Author(s):  
Doron Zeilberger
Keyword(s):  
Analytic Function ◽  
Half Plane ◽  
Analytic Functions ◽  
Complex Analysis ◽  
Discrete Analogue ◽  
Bounded Analytic Functions ◽  
Bounded Analytic Function ◽  
Wiener Theorem ◽  
Upper Half Plane

In this note we prove a discrete analogue to the following Paley–Weiner theorem: Let f be the restriction to the line of a bounded analytic function in the upper half plane; then the spectrum of f is contained in ([0, ∈). The discrete analogue of complex analysis is the theory of discrete analytic functions invented by Lelong-Ferrand (1944) and developed by Duffin (1956) and others.

scholarly journals About the Optimal Recovery of Derivatives of Analytic Functions from Their Values at Points That Form a Regular Polygon

2019 ◽  
pp. 30-38
Author(s):  
Mikhail Ovchintsev
Keyword(s):  
Approximation Method ◽  
Blaschke Product ◽  
Analytic Functions ◽  
Best Approximation ◽  
Extremal Function ◽  
Regular Polygon ◽  
Optimal Recovery ◽  
Bounded Analytic Functions ◽  
The Best Approximation ◽  
Derivatives Of

In this paper, the author solves the problem of optimal recovery of derivatives of bounded analytic functions defined at the zero of the unit circle. Recovery is performed based on information about the values of these functions at points z1, ... , zn , that form a regular polygon. The article consists of an introduction and two sections. The introduction talks about the necessary concepts and results from the works of Osipenko K.Yu. and Khavinson S.Ya., that form the basis for the solution of the problem. In the first section, the author proves some properties of the Blaschke product with zeros at the points z1, ... , zn. After this, the error of the best approximation method of the derivatives f(N)(0), 1 ≤ N ≤ n − 1, by the values f(z1), ... , f(zn) is calculated. In the same section he gives the corresponding extremal function. In the second section, the uniqueness of the linear best approximation method is established, and then its coefficients are calculated. At the end of the article, the formulas found for calculating of the coefficients are substantially simplified.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

scholarly journals Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator

2021 ◽  
Vol 19 (1) ◽  
pp. 329-337
Author(s):  
Huo Tang ◽  
Kaliappan Vijaya ◽  
Gangadharan Murugusundaramoorthy ◽  
Srikandan Sivasubramanian
Keyword(s):  
Analytic Function ◽  
Differential Operator ◽  
Lower Bounds ◽  
Analytic Functions ◽  
Function Class ◽  
Partial Sums ◽  
Generalized Function ◽  
Inclusion Relations ◽  
Alpha Beta

Abstract Let f k ( z ) = z + ∑ n = 2 k a n z n {f}_{k}\left(z)=z+{\sum }_{n=2}^{k}{a}_{n}{z}^{n} be the sequence of partial sums of the analytic function f ( z ) = z + ∑ n = 2 ∞ a n z n f\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n} . In this paper, we determine sharp lower bounds for Re { f ( z ) / f k ( z ) } {\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\} , Re { f k ( z ) / f ( z ) } {\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\} , Re { f ′ ( z ) / f k ′ ( z ) } {\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\} and Re { f k ′ ( z ) / f ′ ( z ) } {\rm{Re}}\{{f}_{k}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}^{^{\prime} }\left(z)\} , where f ( z ) f\left(z) belongs to the subclass J p , q m ( μ , α , β ) {{\mathcal{J}}}_{p,q}^{m}\left(\mu ,\alpha ,\beta ) of analytic functions, defined by Sălăgean ( p , q ) \left(p,q) -differential operator. In addition, the inclusion relations involving N δ ( e ) {N}_{\delta }\left(e) of this generalized function class are considered.

scholarly journals On approximation by bounded analytic functions

1954 ◽  
Vol 5 (1-3) ◽  
pp. 191-196
Author(s):  
J. L. Walsh ◽  
J. P. Evans
Keyword(s):  
Analytic Functions ◽  
Bounded Analytic Functions
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