Generalization of Proximate Order and Applications

We introduce a concept of a quasi proximate order which is a generalization of a proximate order and allows us to study efficiently analytic functions whose order and lower order of growth are different. We prove an existence theorem for a quasi proximate order, i.e. a counterpart of Valiron’s theorem for a proximate order. As applications, we generalize and complement some results of M. Cartwright and C. N. Linden on asymptotic behavior of analytic functions in the unit disc.