L-Index in Joint Variables: Sum and Composition of an Entire Function with a Function With a Vanished Gradient

Abstract

The composition H(z)=f(Φ(z)) is studied, where f is an entire function of a single complex variable and Φ is an entire function of n complex variables with a vanished gradient. Conditions are presented by the function Φ providing boundedness of the L-index in joint variables for the function H, if the function f has bounded l-index for some positive continuous function l and L(z)=l(Φ(z))(max{1,|Φz1′(z)|},…,max{1,|Φzn′(z)|}),z∈Cn. Such a constrained function L allows us to consider a function Φ with a nonempty zero set. The obtained results complement earlier published results with Φ(z)≠0. Also, we study a more general composition H(w)=G(Φ(w)), where G:Cn→C is an entire function of the bounded L-index in joint variables, Φ:Cm→Cn is a vector-valued entire function, and L:Cn→R+n is a continuous function. If the L-index of the function G equals zero, then we construct a function L˜:Cm→R+m such that the function H has bounded L˜-index in the joint variables z1,…,zn. The other group of our results concern a sum of entire functions in several variables. As a general case, a sum of functions with bounded index is not of bounded index. The same is also valid for the multidimensional case. We found simple conditions proving that f1(z1)+f2(z2) belongs to the class of functions having bounded index in joint variables z1,z2. We formulate some open problems based on the deduced results and on the usage of fractional differentiation operators in the theory of functions with bounded index.