  This paper establishes a new Chebyshev criterion for some family of
integrals. By virtue of this criterion we obtain several new Chebyshev
families. With the help of these new families we can answer the conjecture
posed by Gasull et al in 2015. %[J. Differential Equations, 258 (2015),
3286--3303]. Their applications to other two planar differential systems also
show that our approach is simpler and in a unified way to handle many kinds of
planar differential systems for estimating the number of limit cycles
bifurcating from period annulus.
