Let $K$ be a knot in the 3-sphere $S^3$ and $g_1(K)$ the 1-bridge genus of $K$. Concerning the additivity of the 1-bridge genus under connected sum, P. Hoidn showed that $g_1(K_1 \# K_2) \ge g_1(K_1) + g_1(K_2)$ if both $K_1$ and $K_2$ are small. In this talk we generalize the result. In fact we show that $g_1(K_1 \# K_2) \ge g_1(K_1) + g_1(K_2)$ if both $K_1$ and $K_2$ are meridionally small. In addition, we discuss the best possibility on the inequality.