eng Science and Education Publishing Turkish Journal of Analysis and Number Theory 2333-1232 2015-09-28 3 3 87 89 10.12691/tjant-3-3-4 TJANT2015334 article Schur-geometric and Schur-harmonic Convexity of an Integral Mean for Convex Functions Jian Sun 1 Zhi-Ling Sun 1 Bo-Yan Xi 1 Feng Qi qifeng618@gmail.com 2 3 College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, China In the paper, Schur-geometric and Schur-harmonic convexity of an integral mean for convex functions are established. http://pubs.sciepub.com/tjant/3/3/4/tjant-3-3-4.pdf Schur-convex function Schur-geometrically convex function Schur-harmonically convex function inequality generalized logarithmic mean