Toader-Qi平均与其他二元平均的几个确界

Some sharp bounds for Toader-Qi mean of other bivariate means

研究了Toader-Qi平均TQ(a,b)关于几何平均G(a,b)、对数平均L(a,b)、算术平均A(a,b)和二次平均Q(a,b)若干特殊组合的序关系.运用实分析方法以及第1类Bessel函数的乘积公式,建立若干重要引理,导出了4个关于Toader-Qi平均TQ(a,b)的精确不等式,并获得了特殊情形的结果.

This paper study the order relation of some special combinations of geometric mean G（a,b）,logarithmic mean L（a,b）,arithmetic mean A（a,b）and quadratic mean Q（a,b）for Toader-Qi mean TQ（a,b）.By using the method of real analysis in mathematics and the product formula of the first kind Bessel function,several important lemma are established,and four optimal inequalities for Toader-Qi mean TQ（a,b）are found.The results of particular cases are also presented.