Let f(z) be a function analytic in the complex domain C+∪C−∪(−1,+∞), real for real z and such that f(z), f2(z) are anti-Herglotz functions. In this paper we show that f(z) possesses simple upper and lower bounds given by rational functions. Our main tool is the analytic theory of continued fractions. An application to the theory of solutions of the Feigenbaum-Cvitanović equation is given.

We present a simple protocol for two-player secure circuit evaluation. The protocol enables players C and D to cooperate in the computation of f(x) while D conceals her data x from C and C conceals his circuit for f from D. The protocol is based on the technique of hiding information from an oracle [Abadi, Feigenbaum, Kilian, J. Comput. System Sci. 39(1):21{50, August, 1989].

In this paper, we develop a new approach to the construction of solutions of the Feigenbaum-Cvitanović equation whose existence was shown by H. Epstein. Our main tool is the analytic theory of continued fractions.