id author title date pages extension mime words sentence flesch summary cache txt asir-3636 Siddig, Musa; Hussein, Shawgy; Elseid, Amani Validity of Closed Ideals in Algebras of Series of Square Analytic Functions 2021 19 .pdf application/pdf 7475 349 87 |π‘‘πœƒ2 𝑗 2πœ‹ 0 } ≀ exp{ 1 2πœ‹ ∫ βˆ‘ 1βˆ’ (πœ‡(1 βˆ’ πœ–))2 |π‘’π‘–πœƒ 2 βˆ’ πœ‡π‘§| 2 log|𝑓𝑗 2 (π‘’π‘–πœƒ 2 )|π‘‘πœƒ2 𝑗 Λγ } = exp {βˆ’(1 βˆ’ πœ‡(1 βˆ’ πœ–)) infπœƒ2βˆˆΞ›Ξ³ | π‘’π‘–πœƒ 2 βˆ’ 𝑧 π‘’π‘–πœƒ 2 βˆ’ πœ‡π‘§ | 2 π‘Žπ›Ύ(𝑧)}. βˆ’ 𝑓𝑗 2 (πœ‡ (1βˆ’πœ–)𝑒𝑖𝑑 2(1 βˆ’ πœ–)𝑒𝑖𝑑 2 )| πœ–+2 (1 βˆ’ πœ‡ (1βˆ’πœ–)𝑒𝑖𝑑 2) 2 (1 𝑗 𝛾(1βˆ’πœ–)β‹‚βˆ†πœΈ 𝟐𝟐 βˆ’ πœ–)𝑑𝑑2 ≀ 𝐢1+πœ– πœ–1βˆ’πœ€(1+πœ–) ∫ (∫ βˆ‘|(𝑓𝑗 2 )β€² (( 1 2 +πœ–)𝑒𝑖𝑑 2 )| 𝑗 2 𝑑( 1 2 +πœ–) (1βˆ’πœ–) πœ‡ (1βˆ’πœ–)𝑒𝑖𝑑 2(1βˆ’πœ–) )(1 𝛾(1βˆ’πœ–)β‹‚βˆ†πœΈ 𝟐𝟐 βˆ’ πœ–)𝑑𝑑2 ≀ 𝐢1+πœ– πœ–1βˆ’πœ€(1+πœ–) ∫ βˆ‘|(𝑓𝑗 2 )β€² (( 1 2 +πœ–)𝑒𝑖𝑑 2 )| 𝑗 2 ( 1 2 +πœ–) 𝑑( 1 2 +πœ–) 𝑑𝑑2 ( 1 2+πœ–) (1βˆ’πœ–) ≀ 𝐢1+πœ– πœ–1βˆ’πœ€(1+πœ–) ∫ βˆ‘|(𝑓𝑗 2 )β€²(𝑧 βˆ’ πœ–)| 2 𝑗 𝑑𝐴(𝑧 βˆ’ πœ–), ( 1 2+πœ–) (1βˆ’πœ–) Where 𝑆(1βˆ’πœ–) ≔ {(𝑧 βˆ’ πœ–) ∈ 𝔻 ∢ 0 ≀ |𝑧 βˆ’ πœ–| ≀ (1 βˆ’ πœ–) and 𝑧 βˆ’ πœ– |𝑧 βˆ’ πœ–| ∈ 𝛾}. cache/asir-3636.pdf txt/asir-3636.txt