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 conjecture that any planar 3-connected graph can be embedded in the plane in such a way that for any nodes s and t, there is a path from s to t such that the Euclidean distance to t decreases monotonically along the path. A consequence of this conjecture would be that in any ad hoc network containing such a graph as a subgraph, 2-dimensional virtual coordinates for the nodes can be found for...

1. Introduction. A Riemann surface S is called symmetric if there exists an anti-conformal map 0 of S onto itself such that <p2 = identity. We say that 0 is a symmetry on S. The classical "retrospection theorem" asserts the existence of representations of closed Riemann surfaces of genus g by "Schottky groups," groups generated by Möbius transformations Ax,-,Ag such that A; maps the exterior of...

Planar graphs can be represented as the intersection of different types geometric objects in plane, e.g., circles (Koebe, 1936), line segments (Chalopin & Gonçalves, SODA 2009), L-shapes (Gonçalves et al., 2018). For general graphs, however, even deciding whether such representations exist is often NP-hard. We consider apex i.e., that made planar by removing one vertex from them. show, somewhat...

We obtain estimates for derivative and cross–ratio distortion for C (any η > 0) unimodal maps with non–flat critical points. We do not require any ‘Schwarzian–like’ condition. For two intervals J ⊂ T , the cross–ratio is defined as the value B(T, J) := |T ||J | |L||R| where L, R are the left and right connected components of T \J respectively. For an interval map g such that gT : T → R is a dif...

in this study, yield-water and profit function was used to obtain water demand function in agriculture sector. the results showed that the ratio of actual to potential evapotranspiration had positive, significant effecton the ratio of actual to potential yield for wheat and barley. also, square ratio of actual to potential evapotranspiration had negative, significant relationship. after estimat...