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Applications of the theory of differential subordination for functions with fixed initial coefficient to univalent functions

Sumit Nagpal, V. Ravichandran (2012)

Annales Polonici Mathematici

By using the theory of first-order differential subordination for functions with fixed initial coefficient, several well-known results for subclasses of univalent functions are improved by restricting the functions to have fixed second coefficient. The influence of the second coefficient of univalent functions becomes evident in the results obtained.

Bernstein and De Giorgi type problems: new results via a geometric approach

Alberto Farina, Berardino Sciunzi, Enrico Valdinoci (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form div a ( | ∇ u ( x ) | ) ∇ u ( x ) + f ( u ( x ) ) = 0 . Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in  ℝ 2 and  ℝ 3 and of the Bernstein problem on the flatness of minimal area graphs in  ℝ 3 . A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach...

Boundary subordination

Adam Lecko (2012)

Annales Polonici Mathematici

We study the idea of the boundary subordination of two analytic functions. Some basic properties of the boundary subordination are discussed. Applications to classes of univalent functions referring to a boundary point are demonstrated.

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