Complex small entire functions

Small functions were defined in complex analysis together with the notions of order of growth, lower order of growth, type of growth and lower type of growth. The set of small entire functions with respect to an entire function f is a [Formula: see text]-algebra. The set of entire functions with an order of growth strictly inferior to a number t is also a [Formula: see text]-algebra. We then defined very small entire functions with respect to an entire function f and show that if the type of f is finite and the lower type is not 0, then all very small functions with respect to f are small functions but small functions are not always very small functions. Next we consider entire functions [Formula: see text], such that[Formula: see text]Then [Formula: see text], [Formula: see text]. Consider small functions [Formula: see text] with respect to g such that each zero of [Formula: see text] is a zero of [Formula: see text]. Then [Formula: see text]. This applies to very small functions.