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It is shown that if q is a nonnegative continuous function on [O, °°) such that for some positive constants A and L, lim inf fX+ ql/2(t)dt > AL, V_.r^ -7 X X-'OO then y + (r -q)y = 0 has an exponentially increasing solution and an exponentially decreasing solution whenever the uniform norm of the continuous function r satisfies ||r|| < YE/(AL + 1)] . A refinement of the proof is used to show that for all sufficiently large values of k the least eigenvalue X(k) of the two parameter Hill equationdoi:10.1090/s0002-9939-1975-0377184-2 fatcat:eggz6wfoanaxrpijuu2k2ziifa