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The multi-orbital Hubbard model is investigated in order to clarify theelectron correlation effects and the superconductivity in the iron-basedsuperconductor. The renormalization effects on the self-energy and thetwo-particle irreducible vertex function are calculated on the basis of thedynamical mean field theory. We find that the vertex function exhibits a strongrenormalization with the significant orbital dependence as compared with therenormalization factor, when the antiferromagnetic and the antiferro-orbitalfluctuations are comparably enhanced due to the electron and the hole Fermisurfaces nesting effects. The orbital-dependent vertex function together withthe $q\sim0$ nesting between the two-hole Fermi surfaces results in theinter-orbital ferromagnetic fluctuation gradually enhanced which is expected tobe observed in LiFeAs. We show that the hole-$s_{\pm}$-wave superconductivitywith the sign change of the two-hole Fermi surfaces is realized by the enhancedferromagnetic fluctuation accompanied by the antiferromagnetic fluctuation andthe antiferro-orbital fluctuation.

Hole-$s_{\pm}$ State Induced by Coexisting Ferro- and Aniferromagnetic and Aniferro-orbital Fluctuations in the Iron Pnictides