Source 3 is a book sponsered by Cyanotech Corporation claiming beneficial effects for the human health using Astaxanthin. The author is employee of Cyanotech Corporation.
There is a small chapter regarding beneficial effects of Astaxanthin for the tennis elbow, but is is only referring to source 1.

I did not find anymore studies referring to the effects of using Astaxanthin in the treatment of tennis elbow. But there is a [study][1] (sponsered by Cyanotech Corporation) regarding carpal tunnel syndrome claiming a beneficial effect of Astaxanthin while there is another [one][2] stating that there is no significant effect. 

The European Food Saftey Authorizy gave their scientific opinion regarding health claims [here][3]:

> In addition, two unpublished reports on the effects of astaxanthin in the relief of pain and the improvement of performance in patients with carpal tunnel syndrome (Nir and Spiller, 2002a) and with tennis elbow (Spiller et al., 2006), and one intervention in patients with rheumatoid arthritis were 
presented to support the claimed effect (Nir and Spiller, 2002b). The Panel notes that the three studies above are pilot single-centre, double-blind placebo-controlled parallel studies conducted with a product containing high oleic safflower oil, Haematococcus extract, lutein, vitamin A, vitamin E, gelatine and rosemary oil in addition to astaxanthin, and therefore no conclusions can be drawn from these studies on the food that is the subject of the health claim, astaxanthin, in relation to the claimed 
effect. 
The Panel concludes that a cause and effect relationship has not been established between the 
consumption of astaxanthin and the maintenance of normal joints, tendons or connective tissue.


So there is no evidence available supporting the use of Astaxanthin for the treatment of tennis elbow 




  [1]: https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.cyanotech.com/pdfs/bioastin/batl18.pdf&ved=2ahUKEwjBpZz92fvjAhXBwsQBHSu5DM0QFjACegQIARAB&usg=AOvVaw03Lg12cq8x4NKKAxHroM0y&cshid=1565556981743
  [2]: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3280376/#__ffn_sectitle
  [3]: http://www.efsa.europa.eu/en/efsajournal/pub/1253