A Sharp Lower Bound for Toader-Qi Mean with Applications. (17th January 2016)

• Record Type: Journal Article
• Authors: Yang, Zhen-Hang; Chu, Yu-Ming
• Other Names: Zhu Kehe Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2016(2016)
• Publication Date: 2016-01-17
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2016/4165601
• Languages: English
• ISSNs: 2314-8896

Abstract

Abstract : We prove that the inequality T Q ( a, b ) > L p ( a, b ) holds for all a, b > 0 with a ≠ b if and only if p ≤ 3 / 2, where T Q ( a, b ) = 2 / π ∫ 0 π / 2 a c o s 2 θ b s i n 2 θ d θ, L p ( a, b ) = [ ( b p - a p ) / ( p ( b - a ) ) ] 1 / p ( p ≠ 0 ), and L 0 ( a, b ) = a b are, respectively, the Toader-Qi and p -order logarithmic means of a and b . As applications, we find two fine inequalities chains for certain bivariate means.