sup+inf for Riemannian surfaces and sup*inf for for bounded domains of   R^n, n>2
  This paper is in relation with a Note of "Comptes Rendus de l'Academie des Sciences" 2005. We have an idea about a lower bounds of sup+inf (2 dimensions) and sup*inf (dimensions >2). 
Samy Skander Bahoura (IMJ)