References to Liouville go back to his 1847 result that a doubly periodic function without poles is identically constant, which does not yet contain the generalization to either harmonic functions or holomorphic functions.

I quote from <A HREF="https://books.google.nl/books?id=YkUACwAAQBAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false">Barry Simon, Harmonic Analysis: A Comprehensive Course in Analysis, Part 3</A> (page 197):

> That any positive harmonic function is constant is due to Bôcher
> (1902), although the theorem is often named after Picard’s rediscovery
> (1923) — there is often reference to the Liouville–Picard theorem.

Bôcher states the theorem in a footnote:

<IMG SRC="https://ilorentz.org/beenakker/MO/Liouville_1.png"/>    

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 - J. Liouville, <A HREF="https://archive.vn/20120711004552/http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=266004">*Le&ccedil;ons
   sur les fonctions doublement périodiques*</A> (1847).  
   
 -  M. Bôcher, <A HREF="https://projecteuclid.org/euclid.bams/1183417472">*Singular
   points of functions which satisfy partial differential equations of
   the elliptic type*,</A> Bull. Amer. Math. Soc. (2) 9, (1903),
   455–465. 
   
 - É. Picard, *Deux théorèmes élèmentaires sur les singularités des
   fonctions harmoniques*, C. R. Acad. Sci. Paris 176 (1923), 933–935.

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