We consider real interpolation methods defined by means of slowly varying functions$b$and symmetric spaces$E$, for which we present extreme reiteration theorems. As an application we identify, for all possible values of$θ$∈[0,1], the interpolation spaces ($L$_{1},$L$log$L$)_{$θ$,$b$,$E$}and ($L$_{exp},$L$_{∞})_{$θ$,$b$,$E$}.

Asymptotically sharp Bernstein- and Markov-type inequalities are established for rational functions on C2 smooth Jordan curves and arcs. The results are formulated in terms of the normal derivatives of certain Green’s functions with poles at the poles of the rational functions in question. As a special case (when all the poles are at infinity) the corresponding results for polynomials are recaptured.

We study the asymptotic equidistribution of points with discrete energy close to Robin’s constant of a compact set in the plane. Our main tools are the energy estimates from potential theory. We also consider the quantitative aspects of this equidistribution. Applications include estimates of growth for the Fekete and Leja polynomials associated with large classes of compact sets, convergence rates of the discrete energy approximations to Robin’s constant, and problems on the means of zeros of polynomials with integer coefficients.

In this paper, we prove a sum set estimate and the exact sum set theorem for unimodular Kac algebras. Combining the characterization of minimizers of the Donoso–Stark uncertainty principle and the Hirschman–Beckner uncertainty principle, we characterize the extremal pairs of Young’s inequality and extremal operators of the Hausdorff–Young inequality for unimodular Kac algebras.

In this paper we consider the problem of the non-empty intersection of exposed faces in a Banach space. We find a sufficient condition to assure that the non-empty intersection of exposed faces is an exposed face. This condition involves the concept of$inner point$. Finally, we also prove that every minimal face of the unit ball must be an extreme point and show that this is not the case at all for minimal exposed faces since we prove that every Banach space with dimension greater than or equal to 2 can be equivalently renormed to have a non-singleton, minimal exposed face.