Inequalities involving the mean and the standard deviation of nonnegative real numbers

• Main Author: Rojo Oscar
• Format: Article
• Language: English
• Published: SpringerOpen 2006-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2006/43465

<p/> <p>Let <inline-formula><graphic file="1029-242X-2006-43465-i1.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2006-43465-i2.gif"/></inline-formula> be the mean and the standard deviation of the components of the vector <inline-formula><graphic file="1029-242X-2006-43465-i3.gif"/></inline-formula>, where <inline-formula><graphic file="1029-242X-2006-43465-i4.gif"/></inline-formula> with <inline-formula><graphic file="1029-242X-2006-43465-i5.gif"/></inline-formula> a positive integer. Here, we prove that if <inline-formula><graphic file="1029-242X-2006-43465-i6.gif"/></inline-formula>, then <inline-formula><graphic file="1029-242X-2006-43465-i7.gif"/></inline-formula> for <inline-formula><graphic file="1029-242X-2006-43465-i8.gif"/></inline-formula>. The equality holds if and only if the <inline-formula><graphic file="1029-242X-2006-43465-i9.gif"/></inline-formula> largest components of <inline-formula><graphic file="1029-242X-2006-43465-i10.gif"/></inline-formula> are equal. It follows that <inline-formula><graphic file="1029-242X-2006-43465-i11.gif"/></inline-formula> is a strictly increasing sequence converging to <inline-formula><graphic file="1029-242X-2006-43465-i12.gif"/></inline-formula>, the largest component of <inline-formula><graphic file="1029-242X-2006-43465-i13.gif"/></inline-formula>, except if the <inline-formula><graphic file="1029-242X-2006-43465-i14.gif"/></inline-formula> largest components of <inline-formula><graphic file="1029-242X-2006-43465-i15.gif"/></inline-formula> are equal. In this case, <inline-formula><graphic file="1029-242X-2006-43465-i16.gif"/></inline-formula> for all <inline-formula><graphic file="1029-242X-2006-43465-i17.gif"/></inline-formula>.</p>