Abstract. In this paper we show that (K, K )-quasiconformal mappings with unbounded image domains are not Hölder continuous, which is different from the case with bounded image domains given by Kalaj and Mateljević. For a (K, K )-quasiconformal harmonic mapping of the upper half plane onto itself, we prove that it is Lipschitz and hyperbolically Lipschitz continuous. Moreover, we get four equivalent conditions for a harmonic mapping of the upper half plane onto itself to be a (K, K )-quasiconformal mapping.
The main result of this paper is the sharp generalized Schwarz-Pick inequality for euclidean harmonic quasiconformal mappings with convex ranges, which generalizes a result given by Mateljević. As its applications, we obtain the property of quasi-isometry with respect to the Poincaré distance and an analogue of the Koebe theorem for this class of mappings.
Fibrous
surfaces in nature have already exhibited excellent functions
that are normally ascribed to the synergistic effects of special structures
and material properties. The honey bee tongue, foraging liquid food
in nature, has a unique segmented surface covered with dense hairs.
Since honey bees are capable of using their tongue to adapt to possibly
the broadest range of feeding environments to exploit every possible
source of liquids, the surface properties of the tongue, especially
the covering hairs, would likely represent an evolutionary optimization.
In this paper, we show that their tongue hairs are stiff and hydrophobic,
the latter of which is highly unexpected as the structure is designed
for liquid capturing. We found that such hydrophobicity can prevent
those stiff hairs from being adhered to the soft tongue surface, which
could significantly enhance the deformability of the tongue when honey
bees feed at various surfaces and promote their adaptability to different
environments. These findings bridge the relationship between surface
wettability and structural characteristics, which may shed new light
on designing flexible microstructured fiber systems to transport viscous
liquids.
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