A lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons

In this paper, we shall give a lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons. Our result may be regarded as a generalization to Ricci-harmonic geometry of the recent works by Fern'andez-L'opez and Garc'ia-R'io (Q. J. Math. 61, 319–327, 2010), Futaki and Sano (Asian J. Math. 17, 17–32, 2013), and Futaki $\mathit{\text{et}}\mathit{\text{al}}$ (Ann. Global Anal. Geom. 44, 105–114, 2013).