OPTIMAL BOUNDS FOR NEUMAN-SANDOR MEAN IN TERMS OF THE CONVEX COMBINATION OF LOGARITHMIC AND QUADRATIC OR CONTRA-HARMONIC MEANS

In this article, we present the least values alpha(1), alpha(2), and the greatest values beta(1), beta(2) such that the double inequalities alpha L-1(a,b) + 1 (1 - alpha(1))Q(a,b) < M(a,b) < beta L-1(a,b) + (1 - beta(1))Q(a,b) alpha L-2(a,b) + (1 - alpha(2))C(a,b) < M(a,b) < beta L-2(a,b) + (1 - beta(2))C(a,b) hold for all a, b > 0 with a not equal b, where L(a,b), M(a,b), Q(a,b) and C(a,b) are respectively the logarithmic, Neuman-Sandor, quadratic and contra-harmonic means of a and b.