Theorem 2 (p. 212): with z_1 = ... = z_m = 1, the first n-m+1 power sums reach modulus above m(1/2 + m/(8n) + 3m^2/(64n^2))

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Statement

For complex numbers $z_1,\ldots,z_n$ write $S_j=\sum_{t=1}^n z_t^j$ for $j=1,2,\ldots$ (p. 209).

Theorem 2 (p. 212, quoted). "Let $m$ be a positive integer and assume that (7) $z_1=z_2=\ldots=z_m=1$. For arbitrary $n>m$ and every system $z_1,z_2,\ldots,z_n$ satisfying (7) we have"

$$\max_{1\leq j\leq n-m+1}|S_j|>m\left(\frac12+\frac18\frac mn+\frac3{64}\left(\frac mn\right)^2\right).$$

So the range of indices shrinks to $1\leq j\leq n-m+1$ as the number $m$ of prescribed ones grows. At $m=1$ (and $n\geq2$) it reads

$$\max_{1\leq j\leq n}|S_j|>\frac12+\frac1{8n}+\frac3{64n^2},$$

the form in which the introduction (p. 210) calls Theorem 2 "a more precise form of Theorem 1"; the introduction adds that the case of several ones explains why near-extremal systems with more ones are not worth seeking.

Remarks in the paper (pp. 215--216)

• Remark 1 (p. 215). Choosing the angular parameter of the proof by $\cos^2\alpha=1/(1+\sqrt{1-m/n})$ instead of $\alpha=\pi/4$ improves the coefficient of $(m/n)^2$ in Theorem 2 from $3/64$ to $1/16$. The paper states this without writing out the computation.
• Remark 2 (p. 215). For systems satisfying (7), arbitrary $n>m$ and $0<\alpha<\pi/2$, if $\max_{1\leq j\leq n-m}|S_j|\leq m\sin2\alpha/2$, then $|S_{n-m+1}|>m\cos\alpha$; at $\alpha=\pi/4$, a maximum at most $m/2$ over the first $n-m$ power sums forces $|S_{n-m+1}|>m/\sqrt2$.
• Remark 3 (pp. 215--216). For $m=1$ the paper outlines a further improvement and states $R_n>\frac12+\frac{0.159}n$ for sufficiently large $n$, where $R_n$ is the minimum of $\max_{1\leq j\leq n}|S_j|$ under $\max_t|z_t|=1$. The argument is given only in outline: the constant $0.159$ is asserted after "the above geometric arguments" without its computation.

Proof pointer

Pp. 212--215. The proof follows the pattern of Theorem 1, with the polynomial now built on the roots $z_{m+1},\ldots,z_n$. Since the power sums of those roots are $S_j-m$, Newton--Girard gives the paper's (8) and (9), in which the coefficient partial sums appear multiplied by $m$. Lemma 2 (p. 212) is a sharpened planar dichotomy for a nonzero complex $z$ and a parameter $A>0$, and Lemma 3 (p. 213) applies it with $A=k/m$ to obtain, for each $k\leq n-m$, either a large Newton--Girard right-hand side (with the extra factor $1+\cos^2\alpha/(n/m-\cos^2\alpha)$, which uses $k\leq n-m$) or growth of the partial sums; the card states both lemmas. The two cases, run as in Theorem 1, give the lower bounds $m\cos\alpha$ (the paper's (14)) and

$$m\sin\alpha\cos\alpha\sqrt{1+\frac{\cos^2\alpha}{n/m-\cos^2\alpha}}$$

(the paper's (15)). At $\alpha=\pi/4$ the smaller of the two is the second, and expanding $(1-m/(2n))^{-1/2}$ by the binomial series to second order gives the stated bound.

Depends on. Lemmas 2 and 3 of the paper, whose statements are recorded on the [[analysis/biro_1994_problem_turan_concerning_sums_powers_complex/_index|source card]]; the iterated growth estimate is the same as in Lemma 1.

Source. András Biró, On a problem of Turán concerning sums of powers of complex numbers, Acta Math. Hungar. 65 (1994), no. 3, 209--216, doi:10.1007/BF01875148: the statement on printed p. 212, the proof on pp. 212--215, Remarks 1--3 on pp. 215--216.

Read depth. Claims checked: the statement, the two lemmas it uses and the three remarks were read clause by clause on the printed pages. The proof was read but not checked step by step, and Remark 1's optimization and Remark 3's constant were not recomputed. Nothing here is independently reviewed.

Bears on

• Problem 519: the problem asks for an absolute $c>0$ with $\max_{1\leq k\leq n}|\sum_iz_i^k|>c$ whenever $z_1=1$. The case $m=1$ of Theorem 2 gives, for every $n\geq2$, the bound $\frac12+\frac1{8n}+\frac3{64n^2}$, which exceeds Theorem 1's $\frac12$ by a term that tends to $0$; it gives no absolute constant above $\frac12$.