On a problem of H. Shapiro Original Research Article

Let μ be a real measure on the line such that its Poisson integral M(z) converges and satisfies|M(x+iy)|⩽Ae−cyα, y→+∞,for some constants A,c>0 and 0<α⩽1. We show that for 1/2<α⩽1 the measure μ must have many sign changes on both positive and negative rays. For 0<α⩽1/2 this is true for at least one of the rays, and not always true for both rays. Asymptotical bounds for the number of sign changes are given which are sharp in some sense.