arXiv:1908.01696v2 [math-ph] 13 Oct 2019 Elements of Generalized Tsallis Relative Entropy in Classical Information Theory Supriyo Dutta 1* , Shigeru Furuichi 2 † , Partha Guha 1 ‡ 1 Department of Theoretical Science S. N. Bose National Centre for Theoretical Sciences Block - JD, Sector - III, Salt Lake City, Kolkata West Bengal, India - 700 106 2 Department of Information Science, College of Humanities and Sciences, Nihon University, 3-25-40, Sakurajyousui, Setagaya-Ku, Tokyo, 156-8550, Japan Abstract This article proposes a modification in the Sharma-Mittal entropy and distinguishes it as generalised Tsallis entropy. This modification accom- plish the Sharma-Mittal entropy to be used in classical information theory. We derive a product rule (xy) r+k ln {k,r} (xy)= x r+k ln {k,r} (x)+ y r+k ln {k,r} (y)+2kx r+k y r+k ln {k,r} (x) ln {k,r} (y), for the two-parameter deformed logarithm ln {k,r} (x)= x rx k -x -k 2k . It as- sists us to derive a number of information theoretic properties of the gener- alized Tsallis entropy, and related entropy. They include the sub-additive property, strong sub-additive property, joint convexity, and information monotonicity. This article is an exposit investigation on the information- theoretic, and information-geometric characteristics of generalized Tsallis entropy. 1 Introduction Information geometry [1] has been developed in the field of statistics as a geo- metric way to analyse different order dependencies between random variables. The information geometry has a unique feature. It has a dualistic structures of affine connections. In this article, we study information geometry of a two * Email: dosupriyo@gmail.com † Email: furucihi@chs.nihon-u.ac.jp ‡ Email: partha@bose.res.in 1