difference of probability over two different surfaces

hi,
can anybody help me with this problem?
say, i have a continuous probability density f(x). now i have two
different surfaces S1,S2 and i want to estimate the difference (some
lower bound) between probablity over these two surfaces, i.e absolute
value of integrate f(x) over S1 minus integrate f(x) over S2.

when i know that f(x) comes from a mixture of two distributions f1,f2,
such that f(x)=p*f1(x)+(1-p)*f2(x), where p is the prior probablity of
the first distribution , and S1, S2 are the solution of two different
equations involving f1,f2, how do i get an estimate ( lower bound) of
the difference mentioned above?

thanks

kaushik

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