Sharp Geometric Mean Bounds for Neuman Means

Abstract

We find the best possible constants ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2\in [\mathrm{0,1}/2]$ and ${\alpha }_3,{\alpha }_4,{\beta }_3,{\beta }_4\in [1/\mathrm{2,1}]$ such that the double inequalities $G({\alpha }_{1}a+(1-{\alpha }_1)b,{\alpha }_{1}b$ + + $(1-{\beta }_1)b,{\beta }_{1}b+(1-{\beta }_1)a),$ $G({\alpha }_{2}a+(1-{\alpha }_2)b,{\alpha }_{2}b$ + + $(1-{\beta }_2)b,{\beta }_{2}b+(1-{\beta }_2)a),$ $Q({\alpha }_{3}a+(1-{\alpha }_3)b,{\alpha }_{3}b$ + + $(1-{\beta }_3)b,{\beta }_{3}b+(1-{\beta }_3)a),$ $Q({\alpha }_{4}a+(1-{\alpha }_4)b,{\alpha }_{4}b$ + + $(1-{\beta }_4)b,{\beta }_{4}b+(1-{\beta }_4)a)$ hold for all $a,b>0$ with $a\ne b$, where $G$, $A$, and $Q$ are, respectively, the geometric, arithmetic, and quadratic means and $N_{AG}$, $N_{GA}$, $N_{QA}$, and $N_{AQ}$ are the Neuman means.