Log-convex solutions of the second order to the functional equation f(x+1)=g(x)f(x)f(x+1)=g(x)f(x)

In this paper we discuss log-convex solutions of the second order f:R+→R+ to the functional equation with initial condition given byequation(E)f(x+1)=g(x)f(x)for all x∈R+,f(1)=1. We prove that if g satisfies an appropriate asymptotic condition, then (E) admits at most one solution f, which is eventually log-convex of the second order. Moreover, f can be defined explicitly in terms of g. If, in addition, g is eventually log-concave of the second order, then (E) has exactly one eventually log-convex of the second order solution. Our results complement similar ones established by R. Webster [R. Webster, Log-convex solutions to the functional equation f(x+1)=g(x)f(x)f(x+1)=g(x)f(x): Γ-type functions, J. Math. Anal. Appl. 209 (1997) 605–623] and generalize results obtained by L. Lupaş [L. Lupaş, The C-function of E.W. Barnes, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 1 (1990) 5–11].