Abstract
A. A. Goldberg in Possible Magnitude of the Lower Order of an Entire Function with a Finite Deficient Value [4] poses the question of existence of entire functions of infinite order, finite lower order, and having a finite deficient value. The answer to both questions is affirmative. We prove existence by constructing an explicit infinite product representation of an entire function with zero having positive deficiency to meet the requirements. Our methods include generalizing a result of B. Ja. Levin concerning particular entire functions with zeros evenly distributed on two rays. Next we exhibit a polynomial substitute for the exponential convergence factor which appears in the standard Weierstrass primary factor. Then we partition the complex plane into annular regions which are appropriate for our purposes of interpolating through a family of entire functions. Finally we take a comparison function by D. Drasin and generalize it to obtain a counting function for zeros which we use to construct our entire function.
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