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...

We discuss how the statistical properties of the area and radius of gyration of single self-avoiding loops, and of Ising and percolation cluster boundaries, may be calculated using ideas of two-dimensional field theory. For cluster boundaries, we show that almost all loops have area C lnL+O(1), where L is the size of the system, and C is a calculable constant. We also compute the universal rati...

We make a survey of recent results on the geometrical properties of smooth bump functions and the geometrical and topological properties of starlike bodies in Banach spaces. We try to stress the interplay between infinitedimensional topology and nonlinear functional analysis by relating questions about topological and geometrical properties of starlike bodies to other interesting problems on no...