Differences and similarities between multiplicatively and additively universal entire functions and universality properties of compositionally non-normal families of holomorphic functions

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The objective of the research project is to develop differences and similarities between multiplicatively and additively universal entire functions and to analyse universality properties of compositionally non-normal families.In the first phase of the planned project, additional properties of multiplicatively and additively universal entire functions shall be elaborated. Within this research area, there already exist a number of results in the literature concerning Julia directions, boundedness and exponential decay of universal functions. However, the question whether multiplicatively universal entire functions, analogously to additively universal entire functions, may have arbitrarily small transcendental growth or if these functions are subject to certain growth restrictions on the whole plane is still open. The main subject of the first phase of the project is to answer this research question. As the Riemann zeta function fulfils a certain additive universality property on the right half of the critical strip and, if the Riemann hypothesis is true, has no zeros there, it furthermore shall be clarified if multiplicatively universal functions may also be zero-free.In the second phase of the project, it shall be investigated in how far compositionally non-normal family of holomorphic functions allow universality. The existence of a compositionally universal function for a normal family of holomorphic functions implies that the corresponding family of post-compositions is not a normal family any more. However, the reverse implication does not hold in general; the non-normal family consisting of the exponential function which is scaled in the argument with positive integers serves as a motivating example here. Therefore, the question arises which degree of universality a compositionally non-normal family of holomorphic functions can have. An answer to this research question, by again distinguishing possible differences and similarities between multiplicative and additive universality, shall be developed in the second phase of the project.As a result of the successful implementation of the research project, I expect a significant scientific development in a new mathematical environment.