Abstract
Several years ago, E. Landau2 proved the following interesting theorem:
If one considers the set E of all power series
$$ f\left( x \right) = {c_0} + {c_1}x + {c_2}{x^2}... $$converging for |x| < 1 which satisfy the condition M(f) ≤ 1 then the upper limit Gn of the expression |c0+c1+... +cn| is
$$ {G_n} = 1 + {\left( {\frac{1}{2}} \right)^2} + {\left( {\frac{{1.3}}{{2.4}}} \right)^2} + ... + {\left( {\frac{{1.2...\left( {2n - 1} \right)}}{{2.4...2n}}} \right)^2} $$for every value of n.
This paper first appeared in German in “Journal für Reine und Angewandte Mathematik,” Vol. 148 (1918) pp. 122–145.
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References
Berlin Journal für Mathematik, Vol.147, 1917, p.205–232. In the following text, P.I refers to Part I of this article.
Archiv d. Math. u. Phys., Vol.21, 1913, p.250. Cf. also E. Landau, Darstelling und Begruendung einiger neuerer Ergebnisse der Functionentheorie, Berlin, 1916, p.20.
For the case R(f(x)) > 0 a similar remark can be found (due to O. Toeplitz) in E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, p.891. Cf. also F. Riesz, Berlin Journal fuer Mathematik, Vol.146, 1915, p.85.
If the real part of ∑b ν xν is positive for |x| < 1 then we clearly have |b ν | ≤ 2b0′ for ν ≥ 1. This result follows immediately from the inequality (35) if we set f(x) = xν.
Muench. Ber., No.3, 1910. E. Landau gives a somewhat simpler example in Section 3 of his book cited in Footnote 2.
Cf. my article in Berlin Journal für Mathematik, Vol.140, 1911, p.20.
Cf. L. Fejér, Berlin Journal für Mathematik, Vol.146, 1915, p.59.
Bull. de l’Acad. Sc. de Belgique, 1908, p.193. Cf. also T.H. Gronwall, Berlin Journal fuer Mathematik, Vol.147, 1916, p.16.
Rendiconti di Palermo, Vol.38, 1914, p.95.
Cf. E. Landau, Arch. d. Math. u. Phys., Vol.21, 1913, p.42.
H. Bohr, in Goett. Nachr., math-phys. Kl., 1917, p.119–128, has shown that there exist special functions of the class E, for which (math). (Here Gn denotes Landau’s upper bound for |sn|, mentioned in Section 9.) Since (math) and (math), (math) does not exist in Bohr’s case. Furthermore, Bohr proves elsewhere that (math), for every function of the class E. This follows immediately from the inequality (40′’), proven in the text, combined with the fact that (math). (Note added in proof.)
Acta Math., Vol.30, 1906, p.335.
More generally, f(x) converges towards F(φ) when x approaches the point e1φ along a straight line from the interior of the unit circle; cf. E. Study, Konforme Abbildung einfach-zusammenhaengender Bereiche, Teubner, 1913, p.50.
Note that the numbers a nν are independent of the choice of rn+1, rn+2, ....
Cf. H. Gronwall, Annals of Math., Vol. 14, 1912, p. 72, and W. Blaschke, Leipziger Berichte, Vol. LXVII, 1915, p. 194.
Obviously, the same holds for an arbitrary power series φ (t).
Die Dynamik der Systeme starrer Koerper, (German edition, Liepzig, 1898), Vol.2, Section 291.
Math. Ann., Vol.46, 1895, p. 273. Cf. also L. Orlando, Math. Ann., Vol. 71, 1912, p.233.
G. Pólya, whom I informed of this theorem, kindly drew my attention to yet another proof, due to A. Hurwitz, and based on a theorem of Biehler (Berlin Journal fuer Mathematik, Vol. 87, p. 350).
Werke, Vol.III, 1866, p.112. For further references: L. Fejér, Math. Ann., Vol.65, 1908, p.417.
The formulas in the text can be proven most elegantly by means of matrix calculus. We start from the equation which follows easily from (46). Besides, these are well-known formulas for continued fractions.
However. An (x) can assume the value 1.
This result also follows from the formula which can be proven by means of the equation (4 7).
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Schur, I. (1986). On Power Series Which are Bounded in the Interior of the Unit Circle II. In: Gohberg, I. (eds) I. Schur Methods in Operator Theory and Signal Processing. Operator Theory: Advances and Applications, vol 18. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5483-2_4
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