Optimal bounds for Neuman-Sándor mean in terms of the convex combination of logarithmic and quadratic or contra-harmonic means

摘要

In this article, we present the least values α 1 , α 2 , and the greatest values β 1 , β 2 such that the double inequalities α 1 L(a,b)+(1 . α 1 )Q(a,b) 0 with a = b ,where L(a,b) , M(a,b), Q(a,b) and C(a, b) are respectively the logarithmic, Neuman-S′ andor, quadratic and contra-harmonic means of a and b .