Geodesic
Inequalities on planar curves in the vicinity of one or two vertices
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 7, Tome 280 (2001), pp. 194-210

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An arc of a plane curve which is one-to-one projectable onto its chord is considered. Inequalities between curvatures and tangent angles at boundary points in relation to the number of vertices are established. For one-vertex curves, restrictions on the curves position are given. A generalized notion of vertex a curve is used and applied to the four-vertex theorem. 
@article{ZNSL_2001_280_a13,
     author = {A. I. Kurnosenko},
     title = {Inequalities on planar curves in the vicinity of one or two vertices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {194--210},
     publisher = {mathdoc},
     volume = {280},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_280_a13/}
}
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A. I. Kurnosenko. Inequalities on planar curves in the vicinity of one or two vertices. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 7, Tome 280 (2001), pp. 194-210. http://geodesic.mathdoc.fr/item/ZNSL_2001_280_a13/