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Generalized Logarithmic Derivative Estimates of Gol'dberg–Grinshtein Type
Author(s) -
Heittokangas J.,
Korhonen R.,
Rättyä J.
Publication year - 2004
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609303002649
Subject(s) - meromorphic function , mathematics , logarithm , logarithmic derivative , derivative (finance) , mathematics subject classification , order (exchange) , combinatorics , type (biology) , pure mathematics , mathematical analysis , ecology , finance , financial economics , economics , biology
For f meromorphic in the complex plane and φ meromorphic in the unit disc, sharp upper bounds are obtained form ( r ,f ( k )f ( j )) = 1 2 π∫ 0 2 πlog + |f ( k )( r e i θ)f ( j )( r e i θ)| d θ ,r < ∞ ,andm ( r ,ϕ ( k )ϕ ( j )) = 1 2 π∫ 0 2 πlog + |ϕ ( k )( r e i θ)ϕ ( j )( r e i θ)| d θr < 1 ,where k and j are integers satisfying k > j ⩾ 0. The results generalize the logarithmic derivative estimate due to Gol'dberg and Grinshtein to derivatives of higher order. 2000 Mathematics Subject Classification 30D35.