- EXISTENCE OF A MINIMIZER FOR THE ENTROPY 29
where C = ~ (max.B(p,inj(p)) R - i log(47r) - ri - μ(g, 1)). Moreover, Jensen's
inequality says that if cp : [O, oo) -+ JR is convex, then
Al( r ) Js(p,r) { cp o W^00 dCT 2:: cp (Al( r ) Js(p,r) { W^00 dCT) = cp (F (r)).
Applying Jensen's inequality w.ith cp (x) = xlogx to (17.83), we have
2L ( r) 2:: F ( r) log F ( r).Applying these two inequalities to (17.85), we obtain(17.86) :r (G (r) A (r)) ::; (CF (r) - tF (r) log F (r)) A (r).
Since Woo 2:: 0, Woo (p) = o, and Woo E C^1 •°', there exists a sequence
ri -+ o+ such that
Thus integrating (17.86) on the interval [ri, r] and taking i -+ oo, we haveG ( r) ::; A~ r) for (CF ( s) -1 F ( s) log F ( s)) A ( s) ds.
Substituting this into (17.81) yieldsF' (r) ::; CrF (r) +A ~r) for (CF (s) - tF (s) log F (s)) A (s) ds.
Since limr--to+ F (r) = 0, we obtain
F(t)::;C fotrF(r)dr+ lat Ad[r)for (cF(s)-tF(s)logF(s))A(s)ds
fort E (O,inj (p)).
Now assume that to::; inj (p) is small enough so that fort E (0, to],
(1) C1tn-l::; A (t) ::; C2tn-l, where C1 > 0 and C2 < oo,
(2) 0::; F (t)::; 1.Then fort E (0, to],
F(t)::; C (lat rF(r)dr-lat r~:l for sn-^1 F(s)logF(s) ds)
+C ft d:
1r sn-^1 F(s) ds
Jo rn Jofor some C < oo. Now for a E (0, to] there exists b = b (a) < oo (where
lima--to+ b (a) = 0) such that
F(s)::;b