algorithm - Complexity analysis after multiplication of two functions -

given f(n) = θ(n)

h(n) = o(n)

g(n) = Ω(n)

then order of f(n) + [g(n) . h(n)] ?

edit: f(n) = θ(n) not q(n)

there isn't enough information function p(n) = g(n)*h(n). know g grows at least linearly; growing quadratically, cubically, exponentially. likewise, know h grows at most linearly; growing logarithmically, or constant, or decreasing. result, p(n) decreasing or increasing without bound, means sum f(n) + p(n) decreasing or increasing without bound.

suppose, though, assume h(n) = Ω(1) (i.e., @ least not decreasing). can following p(n):

p(n) = h(n) * g(n)      >= c1 * g(n)      = Ω(g(n)) = Ω(n)  p(n) <= c1*n * g(n)       = o(n*g(n)) 

thus f(n) + p(n) = Ω(n) , f(n) + p(n) = o(n*g(n)), nothing more can said; both bounds tight can make them without more information h or g.