scholarly journals Estimates on derivatives and logarithmic derivatives of holomorphic functions and Picard's theorem

2016 ◽  
Vol 442 (2) ◽  
pp. 446-450 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Holomorphic Functions ◽  
Picard’S Theorem ◽  
Picard's Theorem ◽  
Derivatives Of ◽  
Logarithmic Derivatives

CURVES, DOMAINS AND PICARD'S THEOREM

2002 ◽  
Vol 34 (2) ◽  
pp. 205-211 ◽  
Author(s):  
GUANG YUAN ZHANG
Keyword(s):  
Complex Plane ◽  
Holomorphic Functions ◽  
Geometrical Proof ◽  
Polygonal Curves ◽  
Picard’S Theorem ◽  
Picard's Theorem

Based on computing spherical lengths of polygonal curves and spherical areas of domains bounded by polygonal curves on the complex plane, this paper proves a property of holomorphic functions and, as an application, gives a very brief proof of a famous inequality obtained by L. V. Ahlfors which easily implies Picard's theorem. Taken together with the arguments given by Alhfors, therefore, this paper in fact gives an elementary, brief and geometrical proof of Picard's theorem.

Picard’s Theorem

2001 ◽  
pp. 87-96
Author(s):  
Raghavan Narasimhan ◽  
Yves Nievergelt
Keyword(s):  
Picard’S Theorem ◽  
Picard's Theorem

On Logarithmic Derivatives of Functions in a Class of Starlike Mappings

1993 ◽  
Vol 36 (1) ◽  
pp. 38-44
Author(s):  
Alan D. Gluchoff
Keyword(s):  
Integral Means ◽  
Koebe Function ◽  
Starlike Mappings ◽  
Derivatives Of ◽  
Logarithmic Derivatives

AbstractThe purpose of this paper is to prove some facts about integral means of (d2/dz2)(log[f(z)/z])—or equivalently f″/f, for f in a class of starlike mappings of a "singular" nature. In particular it is noted that the Koebe function is not extremal for the Hardy means Mp(r,f″/f) for functions in this class.

Picard's Theorem Without Tears

1978 ◽  
Vol 85 (4) ◽  
pp. 265-268
Author(s):  
Lawrence Zalcman
Keyword(s):  
Picard’S Theorem ◽  
Picard's Theorem

An Equivalent Form of Picard’s Theorem and Beyond

2018 ◽  
Vol 61 (1) ◽  
pp. 142-148 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Functional Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Equivalent Form ◽  
Entire Solutions ◽  
Partial Differential ◽  
Picard’S Theorem ◽  
Picard's Theorem

AbstractThis paper gives an equivalent form of Picard’s theorem via entire solutions of the functional equation f2 + g2 = 1 and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

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