This paper deals with the second kind of generalized elliptic integral ${\mathcal{E}}_a$, for $a$ in the interval $\left[\frac{1}{2},1\right)$, approximated by the weighted Hölder mean. It establishes sharp bounds of the weighted Hölder mean of ${\mathcal{E}}_a$ in terms of weight, accordingly extending the existing results for the complete case when $a=\frac{1}{2}$ and establishing new inequality relationships.