This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates. 2010 Mathematics Subject Classification. 26D10; 46E35; 58E35. which has provided the opportunity for completing this paper. and we shall denote by f q the L q (−1, 1), dν d norm of

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... . 2) The key to our approach is to combine the ideas of D. Bakry and M. Emery, i.e., take the derivative of i − d e along some flow, with ideas that go back to B. Gidas and J. Spruck in [44] and that were later exploited by M.-F. Bidaut-Véron and L. Véron for getting rigidity results in nonlinear elliptic equations. An unessential but useful trick amounts to write the flow for w with f = w β for some β ∈ R, in the expressions for i and e, as we shall see below. Let us start with the case β = 1. We consider the manifold M p = {w ∈ H : w p = 1} .