Extremality of graph entropy based on degrees of uniform hypergraphs with few edges

Let $mathcal{H}$ be a hypergraph with $n$ vertices. Suppose that $d_1,d_2,ldots,d_n$ are degrees of the vertices of $mathcal{H}$. The $t$-th graph entropy based on degrees of $mathcal{H}$ is defined as $$ I_d^t(mathcal{H}) =-sum_{i=1}^{n}left(frac{d_i^{t}}{sum_{j=1}^{n}d_j^{t}}logfrac{d_i^{t}}{sum_{j=1}^{n}d_j^{t}}right) =logleft(sum_{i=1}^{n}d_i^{t}right)-sum_{i=1}^{n}left(frac{d_i^{t}}{sum_{j=1}^{n}d_j^{t}}log d_i^{t}right), $$ where $t$ is a real number and the logarithm is taken to the base two. In this paper we obtain upper and lower bounds of $I_d^t(mathcal{H})$ for $t=1$, when $mathcal{H}$ is among all uniform supertrees, unicyclic uniform hypergraphs and bicyclic uniform hypergraphs, respectively.