Maximum curves and isolated points of entire functions

Abstract:

Given $M(r,f)=\max _{|{z}|=r}\left ( |{f(z)}|\right )$ , curves belonging to the set of points $\mathcal {M}=\left \{ z:|{f(z)}|=M(|{z}|,f)\right \}$ were defined by Hardy to be maximum curves. Clunie asked the question as to whether the set $\mathcal { M}$ could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions $f_1(z)$ and $f_2(z)$, if the maximum curve of $f_1(z)$ is the real axis, conditions are found so that the real axis is also a maximum curve for the product function $f_1(z)f_2(z)$ . By means of these results an entire function of infinite order is constructed for which the set $\mathcal {M}$ has an infinite number of isolated points. A polynomial is also constructed with an isolated point.