Sharp Bounds for the Weighted Geometric Mean of the First Seiffert and Logarithmic Means in terms of Weighted Generalized Heronian Mean

Abstract

Optimal bounds for the weighted geometric mean of the first Seiffert and logarithmic means by weighted generalized Heronian mean are proved. We answer the question: for $\alpha \in (\mathrm{0,1}),$ what the greatest value $p(\alpha )$ and the least value $q(\alpha )$ such that the double inequality, , holds for all $a,b>\mathrm{0}$ with $a\ne b$ are. Here, $P(a,b),$$L(a,b),$ and $H_{\omega }(a,b)$ denote the first Seiffert, logarithmic, and weighted generalized Heronian means of two positive numbers $a$ and $b,$ respectively.