The authors study the various generalizations of the following variants of the Bernoulli inequality: Let \(0<a\leq 1\leq b,\) \(\varphi (t)=\max \left\{ t^{a},t^{b}\right\} ,\) then for \(c\geq 1\) and all \(t>0,\) \[ \log \left( 1+c\varphi (t)\right) \leq c\max \left\{ \log ^{a}(1+t),b\log (1+t)\right\}. \] The classes of functions which are of logarithmic type for which the counterpart of the above result holds. are studied. In particular, they showed that these classes of functions include the Gaussian hypergeometric functions in the zero-balanced case \(F(a,b;a+b;x).\)