Libera Operator Research Articles

English

We show that if α > 1, then the logarithmically weighted Bergman space $$A_{{{\log }^\alpha }}^2$$ is mapped by the Libera operator L into the space $$A_{{{\log }^{\alpha - 1}}}^2$$ , while if α > 2 and 0 < e ≤ α−2, then the Hilbert matrix operator H maps $$A_{{{\log }^\alpha }}^2$$ into $$A_{{{\log }^{\alpha - 2 - \varepsilon }}}^2$$ . We show that the Libera operator L maps the logarithmically weighted Bloch space $${B_{{{\log }^\alpha }}}$$ , α ∈ R, into itself, while H maps $${B_{{{\log }^\alpha }}}$$ into $${B_{{{\log }^{\alpha + 1}}}}$$ . In Pavlovic’s paper (2016) it is shown that L maps the logarithmically weighted Hardy-Bloch space $$B_{{{\log }^\alpha }}^1$$ , α > 0, into $$B_{{{\log }^{\alpha - 1}}}^1$$ . We show that this result is sharp. We also show that H maps $$B_{{{\log }^\alpha }}^1$$ , α > 0, into $$B_{{{\log }^{\alpha - 1}}}^1$$ and that this result is sharp also.

Hilbert matrix on spaces of Bergman-type

It is well known (see [8,14]) that the Libera operator L is bounded on the Besov space Hνp,q,α if and only if 0<κp,α,ν:=ν−α−1p+1. We prove unexpected results: the Hilbert matrix operator H, as well as the modified Hilbert operator H˜, is bounded on Hνp,q,α if and only if 0<κp,α,ν<1. In particular, H, as well as H˜, is bounded on the Bergman space Ap,α if and only if 1<α+2<p and is bounded on the Dirichlet space Dαp=A1p,α if and only if max⁡{−1,p−2}<α<2p−2. Our results are substantial improvement of [11, Theorem 3.1] and of [6, Theorem 5].

Libera operator on mixed norm spaces hp,q,α ν when 0 &lt; p &lt; 1

Results from [7] on Libera operator acting on mixed norm spaces hp,q,? ?, for 1 ? p ? 1, are extended to the case 0 &lt; p &lt; 1.

Logarithmic Bloch space and its predual

We consider the space B1log?, of analytic functions on the unit disk D, defined by the requirement ?D|f?(z)|?(|z|) dA(z) &lt; ?, where ?(r) = log?(1/(1?r)) and show that it is a predual of the ?log?-Bloch? space and the dual of the corresponding little Bloch space. We prove that a function f(z)=??n=0 an zn with an ? 0 is in B1 log? iff ??n=0 log?(n+2)/(n+1) &lt; ? and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in B1 log?. Some properties of the Cesaro and the Libera operator are considered as well.

Conditions for starlikeness of the Libera operator

Abstract Let "Equation missing" denote the class of functions f that are analytic in the unit disc "Equation missing" and normalized by f ( 0 ) = f ′ ( 0 ) − 1 = 0 . In this paper some conditions are determined for starlikeness of the Libera integral operator F ( z ) = 2 z ∫ 0 z f ( t ) d t . MSC:30C45, 30C80.

Definition and Properties of the Libera Operator on Mixed Norm Spaces

We consider the action of the operator ℒg(z) = (1 − z)−1∫z 1‍f(ζ)dζ on a class of “mixed norm” spaces of analytic functions on the unit disk, X = H α,ν p,q, defined by the requirement g ∈ X⇔r ↦ (1 − r)α M p(r, g (ν)) ∈ L q([0,1], dr/(1 − r)), where 1 ≤ p ≤ ∞, 0 < q ≤ ∞, α > 0, and ν is a nonnegative integer. This class contains Besov spaces, weighted Bergman spaces, Dirichlet type spaces, Hardy-Sobolev spaces, and so forth. The expression ℒg need not be defined for g analytic in the unit disk, even for g ∈ X. A sufficient, but not necessary, condition is that . We identify the indices p, q, α, and ν for which 1°ℒ is well defined on X, 2°ℒ acts from X to X, 3° the implication holds. Assertion 2° extends some known results, due to Siskakis and others, and contains some new ones. As an application of 3° we have a generalization of Bernstein's theorem on absolute convergence of power series that belong to a Hölder class.

Analytic functions with decreasing coefficients and Hardy and Bloch spaces

AbstractThe following rather surprising result is noted.(1) A function f(z) = ∑anzn such that an ↓ 0 (n → ∞) belongs to H1 if and only if ∑(an/(n + 1)) &lt; ∞.A more subtle analysis is needed to prove that assertion (2) remains true if H1 is replaced by the predual, 1(⊂ H1), of the Bloch space. Assertion (1) extends the Hardy–Littlewood theorem, which says the following.(2) f belongs to Hp (1 &lt; p &lt; ∞) if and only if ∑(n + 1)p−2anp &lt; ∞.A new proof of (2) is given and applications of (1) and (2) to the Libera transform of functions with positive coefficients are presented. The fact that the Libera operator does not map H1 to H1 is improved by proving that it does not map 1 into H1.

On the Libera operator

We show that the Libera operator, L , on some spaces of analytic functions is a continuous extension of the conjugate of the Cesàro operator. Results on L acting on various spaces are obtained. In particular, L maps the Bloch space into BMOA. We also prove some results on the best approximation by polynomials in Hardy and Bergman spaces.

The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc

The main results of this note prove that the generalized Libera operator is bounded on the Besov mixed-norm space B α p , q ( D ) as well as on the spaces BMOA and VMOA on the unit disk. The compactness of the operator on B α p , q ( D ) is also studied.

On Libera-type transforms on the unit disc, polydisc and the unit ball

We investigate the boundedness and compactness of the following generalization of the Libera operator where and z 0 belongs to the closure of the unit disk 𝔻. Among other results, it is shown that if p≥1,α≥−1/p and z 0∈∂ 𝔻, the operator is bounded on the mixed norm space if and only if 1/p+(α+1)/q<1. The compactness of the operator is also investigated. We introduce two Libera-type transforms on the unit ball B⊂ℂ n . For one of these operators we give some sufficient conditions to be compact on the mixed norm space on the unit ball, and for the other we show that the operator is bounded on the weighted Bergman space on the unit ball.

Generalized classes of starlike and convex functions of order α

We have introduced, in this paper, the generalized classes of starlike and convex functions of order α by using the fractional calculus. We then proved some subordination theorems, argument theorems, and various results of modified Hadamard product for functions belonging to these classes. We have also established some properties about the generalized Libera operator defined on these classes of functions.