Complex variables

1. Olga Kuznetsova, Vladimir Tkachev, “Length functions of lemniscates”, We study metric and analytic properties of generalized lemniscates E_t(f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E_t(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E_t(f)| is convex on any interval free of critical points of ln|f|. As another application we deduce explicit formulas of the length function in some special cases., Manuscr. Math., 112:4 (2003), 519-538    

• Saratov Socio-Economic Institute
• Volgograd State University