There seems to be an issue with the way the eyes are modeled causing unrealistic shading of the eyeball. I can only guess that the problem is that the glossy transparent outer shell is two dimensional rather than being a solid. If you make the eyeball itself glossy it looks like this:
But if you make the surrounding shell glossy using Fresnel as the mix factor the result is this:
The reason for the "shadow" is that the light from the lamp is mostly getting reflected instead of refracting onto the eyeball, it's just unlit rather than being a real shadow. I don't know what the effect of a two-dimensional surface instead of a solid/liquid layer is but that's all that comes to mind for the difference. Although the specular reflection are correct in the second image the shadow-like effect is clearly incorrect. Using a specular map to make the white part glossy but the iris non-glossy and using a glossy cornea instead of the outer shell would, I think, more accurately model real eyes, although I won't be surprised if the edges around the cornea are tricky.
Please correct me if I've got this wrong, or if not post your thoughts on the best "fix" for this.