Abstract In this article, it is shown that Koebe inner functions and squares of singular Nevanlinna functions, so called atomic density have minimal realizations in reproducing kernel Krein spaces.

introduction after the rapid population growth in the iranian cities, these cities have experienced great changes. the physical structures of the cities are eroded and the managers have no attention to them. many immigrants entered the cities  have dwelled in the inefficient urban textures like in the marginal areas and old textures. thus, the environmental quality of place has been weak. undes...

A variant of the Circle Packing Theorem states that combinatorial class any convex polyhedron contains elements, called Koebe polyhedra, midscribed to unit sphere centered at origin, and these representatives are unique up Mobius transformations sphere. Motivated by this result, various papers investigate problem centering spherical configurations under transformations. In particular, Springbor...

Let \(\psi\) be univalent, we introduce a newly defined class of analytic functions, given by $$\begin{aligned} {\mathcal {F}}(\psi ):= \left\{ f\in {A}}: \left( \frac{zf'(z)}{f(z)}-1\right) \prec \psi (z),\ (0)=0 \right\} . \end{aligned}$$Note that functions in this may not univalent. Further, establish the growth theorem with some geometrical conditions on and obtain Koebe domain related shar...