How to add two hydroxy groups to 1,5-diphenylpentan-3-ol with as high as possible yield?
A few possibilities for direct hydroxylation:


$\ce{SeO2}$ immediately comes to mind as one potential option; it is well known to selectively oxidize allylic positions to yield alcohols and carbonyl compounds, and there appear to be references suggesting it reacts at the benzylic position as well.
In Advanced Organic Chemistry, March notes that cerium(IV) triflate "converts benzylic arenes to benzylic alcohols, although the major product is the ketone when > 15% of water is present." Ref: Laali, K.K.; Herbert, M.; Cushnyr, B.; Bhatt, A.; Terrano, D. J. Chem. Soc., Perkin Trans. 1 2001, 578.
Intriguingly, March also reports a variety of oxidations accomplished enzymatically. I have no experience and very limited knowledge of these reactions, so I'll just give the relevant primary literature references:

Adam, W.; Lukacs, Z.; Saha-Moller, C.R.; Weckerle, B.; Schreier, P. Eur. J. Org. Chem. 2000, 2923.
Hamada, H.; Tanaka, T.; Furuya, T.; Takahata, H.; Nemoto, H. Tetrahedron Lett. 2001, 42, 909.
Adam, W.; Lukacs, Z.; Harmsen, D.; Saha-Moller, C.R.; Schreier, P. J. Org. Chem. 2000, 65, 878.



If those approaches aren't appealing, there are various less direct routes. Radical bromination (with, e.g., NBS under photochemical conditions), followed either by direct substitution with hydroxide, or alternatively elimination followed by subsequent acid-catalyzed hydration, would likely work well. If you need stereoselectivity, there are are numerous epoxidation reactions (that is, after bromination and subsequent elimination) that would likely be appropriate (Sharpless, Jacobsen, Shi, and certainly numerous others). Conveniently, your molecule would already have the allylic alcohol necessary for Sharpless asymmetric epoxidation, should you want to go that route. In any case, it should be possible to open the epoxides at the less hindered carbon with a suitable hydride donor (e.g., LAH) to yield the alcohols after work-up.