On the converse of Abel's theorem.
Masatsugu Tsuji · Journal of the Mathematical Society of Japan · 1953
Concerning the convcrse of Abel's theorem, Hardy and Littlewood proved the following two theorems. THEOREM 1Let $f(x)=\sum a_{n}$ be regular for $|x|<1$ and $f(x)\rightarrow s$ , when $x$ tends to $x=1$ along the real axis.If $a_{\iota}$ are real and $na_{l}\leq K$ $(n=1,2, \cdots),$ then $\lambda a_{l}=s,$ .The original proof of theorem 1 is very complicated.Recently Wielandt3) gave a remarkably simple proof of it.In this paper, I shall simplify somewhat thc originai proof of Theorem 2.First we shall prove a lemma.LEMMA.Let $D$ be $a$ $\backslash \sigma i:' l/yly$ connected domain on the1) Hardy and Littlewood: Tauberian thcorem concerning power series and Dirichlet's series whose coefficient are $P);itive$ .$P\downarrow o(\backslash $ .$I_{\lrcorner}o$ })( $]_{(}\backslash n$ bTath Soc. 13 (1914).2) Hardy and $Littl_{(}\backslash wood$ : $A\mathfrak{l}\backslash e1'\dot{s}t\}_{1}eorc\downarrow\iota)$ and $it eg$, conversg.(1I).I'roc.London Math.Soc.22 (1923).3) Wielandt: Zur Umkehrung des $Abels^{\backslash }c11_{(}\backslash n$ Stetigkeitss $\iota tz\cdot s$ .$M_{tt[1}$ .Zeits.$5G(195,)$ .