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Suppose V{\nu} is the pseudo-variance function of the Cauchy-Stieltjes Kernel(CSK) family K+({\nu}) generated by a non degenerate probability measure {\nu}with support bounded from above. We determine the formula for pseudo-variancefunction (or variance function V{\nu} in case of existence) under booleanadditive convolution power. This formulas is used to identify the relationbetween variance functions under Boolean Bercovici-Bata Bijection betweenprobability measures. We also gives the connection between boolean cumulantsand variance function and we relate boolean cumulants of some probabilitymeasures to Catalan numbers and Fuss Catalan numbers.

Variance function of boolean additive convolution