GENERALIZED LOGARITHMIC DERIVATIVE ESTIMATES OF GOL'DBERG–GRINSHTEIN TYPE

2003/12/23 by Janne Heittokangas, J. HEITTOKANGAS, Risto Korhonen +3 · 4 citations
Mathematics · #Meromorphic and Entire Functions #Analytic and geometric function theory #Holomorphic and Operator Theory

paper · doi:10.1112/s0024609303002649

Abstract

For f meromorphic in the complex plane and φ meromorphic in the unit disc, sharp upper bounds are obtained for m ( r , f ( k ) f ( j ) ) = 1 2 π ∫ 0 2 π log + | f ( k ) ( r e i θ ) f ( j ) ( r e i θ ) | d θ , r < ∞ , and m ( r , ϕ ( k ) ϕ ( j ) ) = 1 2 π ∫ 0 2 π log + | ϕ ( k ) ( r e i θ ) ϕ ( j ) ( r e i θ ) | d θ r < 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.

Cited by

• A note on a famous theorem of Pang and Zalcman
• Relationship between the value-sharing behavior of an entire function and its derivative, and the analytic structure of a nonlinear differential equation
• Partial sharing and cross sharing of entire function with its derivative
• Meromorphic solution of a certain type of algebraic differential equation

Related

• Estimates of the logarithmic derivative near a singular point and applications
• Estimation for the logarithmic partial derivative and Wiman-Valiron theory in several complex variables
• A lemma on the difference quotients
• Transcendental singularities for a meromorphic function with logarithmic derivative of finite lower order
• Explicit estimates for the logarithmic derivative and the reciprocal of the Riemann zeta function