On the reconstruction via Urysohn-Chlodovsky operators

The main first goal of this work is to introduce an Urysohn type Chlodovsky operators defined on positive real axis by using the Urysohn type interpolation of the given function f and bounded on every finite subinterval. The basis used in this construction are the Fréchet and Prenter Density Theorems together with Urysohn type operator values instead of the rational sampling values of the function. Afterwards, we will state some convergence results, which are generalization and extension of the theory of classical interpolation of functions to operators.

Integral representations of Mittag-Leffler function on the positive real axis

The Mittag-Leffler functions appear in many problems associated with fractional calculus. In this paper, we use the methodology for evaluation of the inverse Laplace transform, proposed by M. N. Berberan-Santos, to show that the three-parameter Mittag-Leffler function has similar integral representations on the positive real axis. Some of the integrals are also presented.

Radially Distributed Values and Normal Families

Let L0 and L1 be two distinct rays emanating from the origin and let ${\mathcal{F}}$ be the family of all functions holomorphic in the unit disk ${\mathbb{D}}$ for which all zeros lie on L0 while all 1-points lie on L1. It is shown that ${\mathcal{F}}$ is normal in ${\mathbb{D}}\backslash \{0\}$. The case where L0 is the positive real axis and L1 is the negative real axis is studied in more detail.

STEINER POLYNOMIALS VIA ULTRA-LOGCONCAVE SEQUENCES

We investigate structural properties of the cone of roots of relative Steiner polynomials of convex bodies. We prove that they are closed, monotonous with respect to the dimension, and that they cover the whole upper half-plane, except the positive real axis, when the dimension tends to infinity. In particular, it turns out that relative Steiner polynomials are stable polynomials if and only if the dimension is ≤ 9. Moreover, pairs of convex bodies whose relative Steiner polynomial has a complex root on the boundary of such a cone have to satisfy some Aleksandrov–Fenchel inequality with equality. An essential tool for the proofs of the results is the characterization of Steiner polynomials via ultra-logconcave sequences.