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To get at the average length of a code word replacing a source letter in interval encoding using the code word sets C1 or C2, we need to recall something about concave (some say, concave down) functions which played a role in the proof of Theorem 5.6.3. If h is a concave function defined on an interval, x1, x2,… are points in that interval, and λ1,λ2,… are nonnegative numbers summing to 1, then (In some treatments, this is true by the definition of concave functions. Whether definition or theorem, we take it as a given fact.) If h is continuous as well, then this inequality holds for infinite sums, provided

(a) Show that if text from a zeroth-order source with alphabet S = {s1, …,sm} and relative source frequencies 1,…, fm is encoded by the interval method using C1, then the average length of a code word replacing a source letter is ≤ 1 + 2H(S), where, as usual, H(S) = Use (iv) in problem 3, above, and the fact that log2 is concave.]

(b) Show that if C2 is used in interval encoding, the average number of bits per source letter is [Verify that h(x) = log2(1+log2 x) is concave on [1,) by taking its second derivative. Proceed as in (a).]