Find in Library

<p/> <p>We derive some optimal convex combination bounds related to Seiffert's mean. We find the greatest values <inline-formula> <graphic file="1029-242X-2011-686834-i1.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2011-686834-i2.gif"/></inline-formula> and the least values <inline-formula> <graphic file="1029-242X-2011-686834-i3.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2011-686834-i4.gif"/></inline-formula> such that the double inequalities <inline-formula> <graphic file="1029-242X-2011-686834-i5.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2011-686834-i6.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2011-686834-i7.gif"/></inline-formula> hold for all <inline-formula> <graphic file="1029-242X-2011-686834-i8.gif"/></inline-formula> with <inline-formula> <graphic file="1029-242X-2011-686834-i9.gif"/></inline-formula>. Here, <inline-formula> <graphic file="1029-242X-2011-686834-i10.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2011-686834-i11.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2011-686834-i12.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2011-686834-i13.gif"/></inline-formula> denote the contraharmonic, geometric, harmonic, and Seiffert's means of two positive numbers <inline-formula> <graphic file="1029-242X-2011-686834-i14.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2011-686834-i15.gif"/></inline-formula>, respectively.</p>