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/