Tension between scalar/pseudoscalar new physics contribution to B_s --> mu+ mu- and B --> K mu+ mu-

New physics in the form of scalar/pseudoscalar operators cannot lower the semileptonic branching ratio B(B --> K mu+ mu-) below its standard model value. In addition, we show that the upper bound on the leptonic branching ratio B(B_s --> mu+ mu-) sets a strong constraint on the maximum value of B(B --> K mu+ mu-) in models with multiple Higgs doublets: with the current bound, B(B --> K mu+ mu-) cannot exceed the standard model prediction by more than 2.5%. The conclusions hold true even if the new physics couplings are complex. However these constraints can be used to restrict new physics couplings only if the theoretical and experimental errors in B(B --> K mu+ mu-) are reduced to a few per cent. The constraints become relaxed in a general class of models with scalar/pesudoscalar operators.