Additional results of Zalcman's Pompeiu problem

Summary In this article we continue our study of the following problem posed by Lawrence Zalcman in 1972. LetS be the closed unit square. For eachz in the interior,S ⁰, ofS letS(z) be the largest closed square inS with centroidz, and for each in the interval (0, 1] letS _(z) be the square homothetic toS(z) with linear ratio . Iff is a continuous function such that its integral overS _(z) vanishes for allz in S⁰, is f =0? We show that the answer is yes if 3/4 < 1.