We construct a holomorphic function f in the unit disc, whose derivative belongs to the Hardy class H1 , and the image of the unit circle under z 7→ ∫ z 1 f ′(ζ) dζ ζ is a simple curve, but f is not univalent.

Let φ be an analytic map of unit disk D into itself, consider the composition operator Cφ defined by Cφ(f) = f◦φ whenever f is analytic on D. In this paper, we discuss necessary and sufficient conditions under which a composition operator on a large class of weighted Hardy spaces is a compact.

Using recent characterizations of the compactness of composition operators on HardyOrlicz and Bergman-Orlicz spaces on the ball ([2, 3]), we first show that a composition operator which is compact on every Hardy-Orlicz (or Bergman-Orlicz) space has to be compact on H∞. Then, although it is well-known that a map whose range is contained in some nice Korányi approach region induces a compact comp...

A classical additive basis question is Waring’s problem. It has been extended to integer polynomial and non-integer power sequences. In this paper, we will consider a wider class of functions, namely functions from a Hardy field, and show that they are asymptotic bases.

Let f be a number-theoretic function. A finite set X of natural numbers is called f -large if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding f -largeness condition. We classify those functions f for which the statement PHf is independent of first order (Peano) arithmetic PA. If f is a fixed iteration of the binary lengt...

It has been proposed that the ability to perform joint weak measurements on postselected systems would allow us to study quantum paradoxes. These measurements can investigate the history of those particles that contribute to the paradoxical outcome. Here we experimentally perform weak measurements of joint (i.e., nonlocal) observables. In an implementation of Hardy's paradox, we weakly measure ...

We introduce a new class of conjugations and characterize complex symmetric Toeplitz operators on the Hardy space with respect to those conjugations. Also, we prove that symmetricity uet property are same for certain operators. also discuss analytic Our results extend several known by providing unified ways treating them

This paper discusses the convexity of range Berezin transform. For a bounded operator T acting on reproducing kernel Hilbert space H (on set X), this is B(T):={?Tkˆx,kˆx?H:x?X}, where kˆx normalized for at x?X. Primarily, we focus characterizing class composition operators Hardy unit disk.

We present a unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Pólya Taikov types for multiple closed operators acting on Hilbert space. apply our results establish new the norms powers Laplace-Beltrami compact Riemannian manifolds derive well-known functions defined d-dimensional space in limit case. Other applications include best approximation ...