To get sharp results on weighted integrability of the Fourier transform
of a function in terms of weighted integrability of the function itself
certain conditions on the function should be imposed. In 1972 R.P. Boas
conjectured that the result of such type is true for monotone functions.
In a joint work with S. Tikhonov, Boas' conjecture is proved in affirmative
even for a wider class of functions - the so called general monotone
functions. This new class will be discussed along with certain of its
applications as well as results related to Boas' conjecture.