Geodesic
Complex Tauberian theorems for the one-sided and two-sided Stieltjes transform
Izvestiya. Mathematics , Tome 8 (1974) no. 1, pp. 145-176.

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In the paper we prove one-sided and two-sided complex Tauberian theorems for the Stieltjes transform. Tauberian theorems of the type of Keldysh theorem [1] with a remainder for the one-sided Stieltjes transform in the real domain were studied in [2]–[5]. Theorems of such type in the complex domain have not been studied in detail. Here only one result [6], [7] for the one-sided case is known. 
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M. A. Subkhankulov; F. I. An. Complex Tauberian theorems for the one-sided and two-sided Stieltjes transform. Izvestiya. Mathematics , Tome 8 (1974) no. 1, pp. 145-176. http://geodesic.mathdoc.fr/item/IM2_1974_8_1_a8/

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[4] Subkhankulov M. A., “Nekotorye obschie tauberovy teoremy s ostatochnym chlenom”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 64, 1961, 239–266

[5] Ganelius T., “Tauberian theorems for the Stiltjes transform”, Math. Scand., 14 (1964), 213–219 | MR | Zbl

[6] Malliavin P., “Un theoreme tauberian avec reste la transformee de Stieltijes”, Comptes Rendus de l'Academ. Sciences, Paris, 255 (1962), 2351–2352 | MR | Zbl

[7] Pleijel A., “On a theorem by P. Malliavin”, Israel J. Math., 1 (1963), 166–168 | DOI | MR | Zbl

[8] Gradshtein I. S., Ryzhik I. M., Tablitsa integralov, summ ryadov i proizvedenii, Fizmatgiz, M., 1962