We study mean convergence of multiple ergodic averages, where the iterates arise from smooth functions of polynomial growth that belong to a Hardy field. Our results include all logarithmico-exponential functions of polynomial growth, such as the functions t 3 / 2 , t log ⁡ t t^{3/2}, t\log t and e log ⁡ t e^{\sqrt {\log t}} . We show that if all non-trivial linear combinations of the functions a 1 a_1 , …, a k a_k stay logarithmically away from rational polynomials, then the L 2 L^2 -limit of the ergodic averages 1 N ∑ n = 1 N f 1 ( T ⌊ a 1 ( n ) ⌋ x ) ⋅ ⋯ ⋅ f k ( T ⌊ a k ( n ) ⌋ x ) \frac {1}{N} \sum _{n=1}^{N}f_1(T^{\lfloor {a_1(n)}\rfloor }x)\cdot \dots \cdot f_k(T^{\lfloor {a_k(n)}\rfloor }x) exists and is equal to the product of the integrals of the functions f 1 f_1 , …, f k f_k in ergodic systems, which establishes a conjecture of Frantzikinakis. Under some more general conditions on the functions a 1 a_1 , …, a k a_k , we also find characteristic factors for convergence of the above averages and deduce a convergence result for weak-mixing systems.