A Generalization of the Havrda-Charvat and Tsallis Entropy and Its Axiomatic Characterization

Abstract

In this communication, we characterize a measure of information of types $\alpha$, $\beta$, and $\gamma$ by taking certain axioms parallel to those considered earlier by Havrda and Charvat along with the recursive relation $H_n$($p_1,\dots ,p_n$; $\alpha$, $\beta$, $\gamma$)$-H_{n-1}$ $(p_1+p_2$, $p_3,\dots ,$ $p_n$; $\alpha$, $\beta$, $\gamma$)$=$ ($A_{(\mathrm{\alpha },\mathrm{\gamma })}/$($A_{(\mathrm{\alpha },\mathrm{\gamma })}-A_{(\mathrm{\beta },\mathrm{\gamma })}$))${\left(p_1+p_2\right)}^{\mathrm{\alpha }/\mathrm{\gamma }}{H}_2(p_1/(p_1+p_2)$, $p_2/(p_1+p_2)$; $\alpha$,$\gamma )+$($A_{(\mathrm{\beta },\mathrm{\gamma })}/{\mathrm{(A}}_{(\mathrm{\beta },\mathrm{\gamma })}-A_{(\mathrm{\alpha },\mathrm{\gamma })}$))${(p_1+p_2)}^{(\mathrm{\beta }/\mathrm{\gamma })}{H}_2(p_1/(p_1+p_2)$, $p_2/(p_1+p_2)$; $\gamma$, $\beta$), $\alpha \ne \gamma \ne \beta$, $\alpha$, $\beta$, $\gamma >0$. Some properties of this measure are also studied. This measure includes Shannon’s information measure as a special case.