Comparison of integrals of positive and positive definite functions on different intervals

Wiener's problem (on R) can equivalently be stated the following way. Does there exists a constant K, such that if f is any positive definite function, then the p-th absolute value integral of f on the interval [-2,2] is at most K times the p-th absolute value integral on I=[-1,1]? The answer is well-known: it is to the positive for p an even integer or infinity, and to the negative for other exponents p.
Recently, for use in a paper on number theory, Konyagin and Shteinikov came to the (discrete version of the) question if there exists a constant C, such that for any positive definite and nonnegative function f its integral on [-2,2] is at most C times the integral on [-1,1].
Gorbachev proved that such a constant C exists. We work out several refinements of the result of Gorbachev and give an almost precise description of the respective constant for arbitrary intervals [a-L,a+L].