On F-Algebras Mp (1<p<∞) of Holomorphic Functions
We consider the classes Mp (1<p<∞) of holomorphic functions on the open unit disk 𝔻 in the complex plane. These classes are in fact generalizations of the class M introduced by Kim (1986). The space Mp equipped with the topology given by the metric ρp defined by ρp(f,g)=f-gp=∫02πlogp1+Mf-gθdθ...
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Wiley
2014-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2014/901726 |
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| author | Romeo Meštrović |
| author_facet | Romeo Meštrović |
| author_sort | Romeo Meštrović |
| collection | DOAJ |
| description | We consider the classes Mp (1<p<∞) of holomorphic functions on the open unit disk 𝔻 in the complex plane. These classes are in fact generalizations of the class M introduced by Kim (1986). The space Mp equipped with the topology given by the metric ρp defined by ρp(f,g)=f-gp=∫02πlogp1+Mf-gθdθ/2π1/p, with f,g∈Mp and Mfθ=sup0⩽r<1f(reiθ), becomes an F-space. By a result of Stoll (1977), the Privalov space Np (1<p<∞) with the topology given by the Stoll metric dp is an F-algebra. By using these two facts, we prove that the spaces Mp and Np coincide and have the same topological structure. Consequently, we describe a general form of continuous linear functionals on Mp (with respect to the metric ρp). Furthermore, we give a characterization of bounded subsets of the spaces Mp. Moreover, we give the examples of bounded subsets of Mp that are not relatively compact. |
| format | Article |
| id | doaj-art-59f08b85665540a4aca63ab5ae8b85a8 |
| institution | Kabale University |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | The Scientific World Journal |
| spelling | doaj-art-59f08b85665540a4aca63ab5ae8b85a82025-02-03T01:25:34ZengWileyThe Scientific World Journal2356-61401537-744X2014-01-01201410.1155/2014/901726901726On F-Algebras Mp (1<p<∞) of Holomorphic FunctionsRomeo Meštrović0Maritime Faculty, University of Montenegro, Dobrota 36, 85330 Kotor, MontenegroWe consider the classes Mp (1<p<∞) of holomorphic functions on the open unit disk 𝔻 in the complex plane. These classes are in fact generalizations of the class M introduced by Kim (1986). The space Mp equipped with the topology given by the metric ρp defined by ρp(f,g)=f-gp=∫02πlogp1+Mf-gθdθ/2π1/p, with f,g∈Mp and Mfθ=sup0⩽r<1f(reiθ), becomes an F-space. By a result of Stoll (1977), the Privalov space Np (1<p<∞) with the topology given by the Stoll metric dp is an F-algebra. By using these two facts, we prove that the spaces Mp and Np coincide and have the same topological structure. Consequently, we describe a general form of continuous linear functionals on Mp (with respect to the metric ρp). Furthermore, we give a characterization of bounded subsets of the spaces Mp. Moreover, we give the examples of bounded subsets of Mp that are not relatively compact.http://dx.doi.org/10.1155/2014/901726 |
| spellingShingle | Romeo Meštrović On F-Algebras Mp (1<p<∞) of Holomorphic Functions The Scientific World Journal |
| title | On F-Algebras Mp (1<p<∞) of Holomorphic Functions |
| title_full | On F-Algebras Mp (1<p<∞) of Holomorphic Functions |
| title_fullStr | On F-Algebras Mp (1<p<∞) of Holomorphic Functions |
| title_full_unstemmed | On F-Algebras Mp (1<p<∞) of Holomorphic Functions |
| title_short | On F-Algebras Mp (1<p<∞) of Holomorphic Functions |
| title_sort | on f algebras mp 1 p ∞ of holomorphic functions |
| url | http://dx.doi.org/10.1155/2014/901726 |
| work_keys_str_mv | AT romeomestrovic onfalgebrasmp1pofholomorphicfunctions |