Sharp weighted Hölder mean bounds for the second kind generalized elliptic integral

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.