scholarly journals Variational Formulas and Cocycle solutions for Directed Polymer and Percolation Models

2016 ◽  
Vol 346 (2) ◽  
pp. 741-779 ◽  
Author(s):  
Nicos Georgiou ◽  
Firas Rassoul-Agha ◽  
Timo Seppäläinen
Keyword(s):  
Directed Polymer ◽  
Variational Formulas

Variational Formulas for Higher Affine Mean Curvatures

1988 ◽  
Vol 13 (3-4) ◽  
pp. 318-326 ◽  
Author(s):  
Li An-Min
Keyword(s):  
Variational Formulas

scholarly journals Extremal Problems for Functions Starlike in the Exterior of the Unit Circle

1962 ◽  
Vol 14 ◽  
pp. 540-551 ◽  
Author(s):  
W. C. Royster
Keyword(s):  
Unit Circle ◽  
Analytic Functions ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Simple Pole ◽  
Image Position ◽  
Inverse Functions ◽  
Variational Formulas

Let Σ represent the class of analytic functions(1)which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion(2).In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.

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