Hironobu Urabe

Bio: Hironobu Urabe is an academic researcher from Kyoto University. The author has contributed to research in topics: Entire function & Factorization. The author has an hindex of 3, co-authored 8 publications receiving 27 citations.

Subnormal Solutions of Second Order Linear Differential Equations With Periodic Coefficients

Abstract: We find the form of all subnormal solutions of equation (1.4). Our results generalize and improve a well-known result of Wittich about equation (1.1). Several examples are given. Higher order equations are discussed.

Permutable entire functions and their Julia sets

Abstract: In 1922–23, Julia and Fatou proved that any 2 rational functions f and g of degree at least 2 such that f(g(z)) = g(f(z)), have the same Julia set. Baker then asked whether the result remains true for nonlinear entire functions. In this paper, we shall show that the answer to Baker's question is true for almost all nonlinear entire functions. The method we use is useful for solving functional equations. It actually allows us to find out all the entire functions g which permute with a given f which belongs to a very large class of entire functions.

On the growth of composite entire functions

Abstract: $g$ are entire functions, then the composite function $fog$ is of infinite order unless (a) $f$ is of finite order and $g$ is a polynomial or (b) $f$ is of order zero and $g$ is of finite order. Since then, many results related to this and some further results (e.g. Clunie [2], [3], Edrei and Fuchs [4], Mori [5], Yang and Urabe [6], Yang [7]) have been obtained. Especially, Song and Yang [8] proved that if $fog$ is of finite lower order, then either $f$ is of finite lower order and $g$ is a polynomial, of $f$ is of zero lower order and $g$ is of finite lower order. An interesting problem is that if $\rho(f)=0(\lambda(f)=0)$ , what kind of conditions will ensure $f\circ g$ to be of either finite or infinite (lower) order. It is well known that for any two transcendental entire functions $f,$ $g$

On subnormal solutions of second order linear periodic differential equations

Abstract: In this paper, we investigate the existence and the form of subnormal solution for a class of second order periodic linear differential equations, estimate the growth properties of all solutions, and answer the question raised by Gundersen and Steinbart.