Let 1 < c < 10/9. For large real numbers R > 0, and a small constant η > 0, the inequality holds for many prime triples. This improves work of Kumchev [Acta Arith. 89 (1999)].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa162-2-3,
author = {Roger Baker and Andreas Weingartner},
title = {A ternary Diophantine inequality over primes},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {159-196},
zbl = {1301.11040},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa162-2-3}
}
Roger Baker; Andreas Weingartner. A ternary Diophantine inequality over primes. Acta Arithmetica, Tome 166 (2014) pp. 159-196. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa162-2-3/