In the paper, we clarify some relations between solutions to star-triangle equation via gauge/YBE correspondence. We consider two relation in terms of Euler's gamma function. derive these from reduction certain basic and hyperbolic hypergeometric integral identities. These identities can be interpreted as equality supersymmetric partition functions a specific three-dimensional N=2 dual theories.

In this paper, we prove some inequalities satisfied by the modified degenerate gamma function which was recently introduced. The tools employed include Holder’s inequality, mean value theorem, Hermite–Hadamard’s and Young’s inequality. By parameter variations, established results reduce to corresponding for classical function.