Composition operators and natural metrics in meromorphic function classes $Q_p$

Document Type : Research Paper


Port Said University‎, ‎Faculty of Science‎, ‎Department of Mathematics, Port Said 42521‎, ‎Egypt.


‎In this paper‎, ‎we investigate some results on natural metrics on the $\mu$-normal functions and meromorphic $Q_p$-classes‎. ‎Also‎, ‎these classes are shown to be complete metric spaces with respect to the corresponding metrics‎. ‎Moreover‎, ‎compact composition operators $C_\phi$ and Lipschitz continuous operators acting from $\mu$-normal functions to the meromorphic $Q_p$-classes are characterized by conditions depending only on $\phi.$


Main Subjects

J‎. ‎M‎. ‎Anderson‎, ‎J‎. ‎Clunie and Ch‎. ‎Pommerenke‎, ‎On Bloch functions and normal functions‎, J‎. ‎Reine‎. ‎Angew‎. ‎Math. 270 (1974) 12--37‎.
R‎. ‎Aulaskari and P‎. ‎Lappan‎, ‎Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal‎, ‎136--146‎,
‎Complex Analysis and its Applications‎, ‎Pitman Res‎. ‎Notes Math‎. ‎Ser.‎, ‎305‎, ‎Longman Sci‎. ‎Tech.‎, ‎Harlow‎, ‎1994‎.
F‎. ‎Bagemihl and W‎. ‎Seidel‎, ‎Sequential and continuous limits of meromorphic functions‎, Ann‎. ‎Acad‎. ‎Sci‎. ‎Fenn‎. ‎Math. 280 (1960) 17 pages‎.
C‎. ‎Cowen and B‎. ‎D‎. ‎MacCluer‎, ‎Composition Operators on Spaces of Analytic Functions‎, ‎Studies in Advanced Mathematics‎, ‎CRC Press‎, ‎Boca Raton‎, ‎1995‎.
M‎. ‎Essen and H‎. ‎Wulan‎, ‎On analytic and meromorphic functions and spaces of $Q_K$ type‎, Illinois J‎. ‎Math. 46 (2002)‎, ‎no‎. ‎4‎, ‎1233--1258‎.
A‎. ‎El-Sayed Ahmed and M‎. ‎A‎. ‎Bakhit‎, ‎Composition operators on some holomorphic Banach function spaces‎, Math‎. ‎Scand. 104 (2009)‎, ‎no‎. ‎2‎, ‎275--295‎.
A‎. ‎Kamal and A‎. ‎El-Sayed Ahmed‎, ‎A property of meromorphic functions with Hadamard gaps‎, J‎. ‎Sci‎. ‎Research Essays 8 (2013)‎, ‎no‎. ‎15‎, ‎633--639‎.
M‎. ‎Kotilainen‎, ‎Studies on Composition Operators and Function Spaces‎, ‎Report Series‎, ‎Department of Mathematics‎, ‎University of Joensuu 11‎, ‎Joensuu‎: ‎University of Joensuu‎, ‎Department of Mathematics (Dissertation) 2007‎.
P‎. ‎Lappan and J‎. ‎Xiao‎, Qα# bounded composition maps on normal classes‎, Note Mat. 20 (2000/01)‎, ‎no‎. ‎1‎, ‎65--72‎.
X‎. ‎Li‎, ‎F‎, ‎Perez-Gonzalez and J‎. ‎Rattya‎, ‎Composition operators in hyperbolic $Q$-classes‎, Ann‎. ‎Acad‎. ‎Sci‎. ‎Fenn‎. ‎Math. 31 (2006)‎, ‎no‎. ‎2‎, ‎391--404‎.
S‎. ‎Makhmutov and M‎. ‎Tjani‎, ‎Composition operators on some M\"obius invariant Banach spaces‎, Bull‎. ‎Aust‎. ‎Math‎. ‎Soc. 62 (2000)‎, ‎no‎. ‎1‎, ‎1--19‎.
R‎. ‎A‎. ‎Rashwan‎, ‎A‎. ‎El-Sayed Ahmed and A‎. ‎Kamal‎, ‎Some characterizations of weighted holomorphic Bloch space‎, Eur‎. ‎J‎. ‎Pure Appl‎. ‎Math. 2 (2009)‎, ‎no‎. ‎2‎, ‎250--267‎.
M‎. ‎Tjani‎, ‎Compact composition operators on Besov spaces‎, Trans‎. ‎Amer‎. ‎Math‎. ‎Soc. 355 (2003)‎, ‎no‎. ‎11‎, ‎4683--4698‎.
J‎. ‎Xiao‎, ‎Holomorphic $Q$ Classes‎, ‎Lecture Notes in Mathematics‎, ‎1767‎, ‎Springer‎-
‎Verlag‎, ‎Berlin‎, ‎2001‎.
S‎. ‎Yamashita‎, ‎Criteria for function to be Bloch‎, Bull‎. ‎Aust‎. ‎Math‎. ‎Soc. 21 (1980)‎, ‎no‎. ‎2‎, ‎223--227‎.
J‎. ‎Zhou‎, ‎Composition operators from $B^\alpha$ to $Q_K$ type spaces‎, J‎. ‎Funct‎. ‎Spaces Appl. 6 (2008)‎, ‎no‎. ‎1‎, ‎89--105‎.