TOADER型平均的调和与NEUMAN平均凸组合界

Sharp Bounds of Convex Combination of Harmonic and Neuman Means for Toader-type Mean

研究了TOADER型平均和调和平均与NEUMAN平均凸组合的序关系,运用单调性L HOSPITAL法则,证明了TOADER型平均关于调和与NEUMAN平均凸组合的确界.作为应用,分别给出了第二类完全椭圆积分关于反双曲正弦和反正切函数的两个不等式.所得结果是对一些己知结果的改进.

In this paper,the order relations of Toader type mean with respect to the convex combination of harmonic and Neuman means are studied.By using the monotonicity L Hospital rule,sharp bounds of convex combination of harmonic and Neuman means for Toader type mean are proved.As applications,two inequalities for the inverse hyperbolic sine and inverse tangent function for the second kind of complete elliptic integral are given,respectively.The results are an improvement on some known results.