I am interested in using a result about Newton's method, which basically states that if f is convex on $[a,b]$ and it holds $f(a)<0$ and $f(b)>0$, then the Newton iteration converges to $x^\*\in[a,b]$ with $f(x^\*)=0$ for every starting value $x\_0\in[a,b]$ with $f(x\_0)\geq 0$. 


The proof is not that complicated. Unfortunately, I haven't found this theorem in English literature yet (only in some German [lecture notes](http://www.mia.uni-saarland.de/Teaching/MFI0708/kap52.pdf)). Could you recommend me a book or a paper, where this might be found?


Please note: I have also asked this question on [math.SE](https://math.stackexchange.com/questions/556794/literature-investigating-root-finding-of-convex-functions) (but without any success).



Literature on root finding of convex Functions
[Global convergence of Newton's method on an interval](http://folk.uib.no/ssu029/Pdf_file/Thorlund-Petersen04.pdf), Lars Thorlund-Petersen (2004).



> 
> Global convergence of Newton's method is considered in the strong
>  sense of convergence for any initial value in an interval. The class
>  of functions for which the method converges globally is characterized.
>  This class contains all increasing convex and increasing concave
>  functions as well as sums of such functions on the given interval.
> 
> 
> 


