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We consider a family of CIFSs of the generalized complex continued fractionswith a complex parameter space. We show that for each CIFS of the family, theHausdorff measure of the limit set of the CIFS with respect to the Hausdorffdimen/sion is zero and the packing measure of the limit set of the CIFS withrespect to the Hausdorff dimension is positive (main result). This is a newphenomenon of infinite CIFSs which cannot hold in finite CIFSs. We prove themain result by showing some estimates for the unique conformal measure of eachCIFS of the family and by using some geometric observations.

Hausdorff measures and packing measures of the limit sets of CIFSs of generalized complex continued fractions