Integral means and Dirichlet integral for analytic functions

For normalized analytic functions f in the unit disk, the estimate of the integral means is important in certain problems in fluid dynamics, especially when the functions are non‐vanishing in the punctured unit disk . We consider the problem of finding the extremal function f which maximizes the integral means for f belong to certain classes of analytic functions related to sufficient conditions of univalence. In addition, for certain subclasses of the class of normalized univalent and analytic functions, we solve the extremal problem for the Yamashita functional where denotes the area of the image of under . The first problem was originally discussed by Gromova and Vasil'ev in 2002 while the second by Yamashita in 1990.     

Estimate of the second term in the spectral asymptotic of Cauchy transform

We give the best possible (respecting to order) estimate of the remainder term in spectral asymptotics of the Cauchy operator and the logarithmic potential type operator. Also, we state a conjecture on the exact value of the remainder term.     

Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type

The Hilbert matrix induces a bounded operator on most Hardy and Bergman spaces, as was shown by Diamantopoulos and Siskakis. We generalize this for any Hankel operator on Hardy spaces by using a result of Hollenbeck and Verbitsky on the Riesz projection and also compute the exact value of the norm of the Hilbert matrix. Using a new technique, we determine the norm of the Hilbert matrix on a wide range of Bergman spaces.     

Power controlled soliton stability and steering in lattices with saturable nonlinearity

Dynamical properties of discrete solitons in nonlinear Schr?dinger lattices with saturable nonlinearity are studied in the framework of the one-dimensional discrete Vinetskii-Kukhtarev model. Two stationary strongly localized modes, centered on site (A) and between two neighboring sites (B), are obtained. The associated Peierls-Nabarro potential is bounded and has multiple zeros indicating strong implications on the stability and dynamics of the localized modes. Besides a stable propagation of mode A, a stable propagation of mode B is also possible. The enhanced ability of the large power solitons to move across the lattice is pointed out and numerically verified.