We continue the study of model-independent constraints on the unitary conformal field theories (CFTs) in four dimensions, initiated in [R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, J. High Energy Phys. 12 (2008) 031]. Our main result is an improved upper bound on the dimension Delta of the leading scalar operator appearing in the operator product expansion (OPE) of two identical scalars of dimension d: phi(d)x phi(d)=1+O-Delta+.... In the interval 1 < d < 1.7 this universal bound takes the form Delta < 2+0.7(d-1)(1/2)+2.1(d-1)+0.43(d-1)(3/2). The proof is based on prime principles of CFT: unitarity, crossing symmetry, OPE, and conformal block decomposition. We also discuss possible applications to particle phenomenology and, via a 2D analogue, to string theory.