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Digital notch filters are useful for removing sinusoidal signals with a single frequency or multi-frequencies. For some applications, the sinusoidal signals being processed may change frequencies, and thus the notch filter must also have a variable notch frequency. Such a digital filter with a variable notch-frequency (VNF) is called variable notch-frequency (VNF) digital filter. Generally speaking, a VNF filter has the ability to change its notch frequency continuously, and can remove sinusoidal signals with variable frequencies. To design a recursive VNF filter, guaranteeing its stability is the top-priority issue. In this paper, a recursive VNF filter is designed by utilizing the least Lp-norm criterion, and the stability is guaranteed through adopting the stability-guarantee methodology based on parameter transformations. To perform the parameter transformations, a function that meets a special condition is required, and such a function is called unity-bounded function. In this paper, various unity-bounded functions are developed for the parameter-transformation-based stability guarantee. The unity-bounded functions play a crucial role in the stability guarantee. This paper details how to design a stable VNF filter by incorporating the parameter transformations employing the unity-bounded functions into the Lp-norm minimization of the magnitude response. Theoretically, the stability of the recursive VNF filter resulting from the design is definitely guaranteed. Moreover, since the coefficients of the designed VNF are variable, changing the filter coefficients enables the user to get a VNF filter with updated notch frequency. [ABSTRACT FROM AUTHOR]