FURTHER IMPROVEMENTS FOR YOUNG'S INEQUALITIES ON THE ARITHMETIC, GEOMETRIC, AND HARMONIC MEAN

In this paper, we obtain some improvements and generalizations of Young's inequali- ties on the arithmetic, geometric, and harmonic mean. For example,<br /> (1) If 0 < a <b, ss>1 and 0 < v<t<1, then<br /> (a del(v )b)(beta)-(a(#v ),b)(beta)/(a del(vt)b)(beta)-(a(# ),b)(beta )<= v(1-v)/<br /> (aV;b)ss - (a double dagger rb)ss t(1 - t) (2) If 0<b<a, beta > 1 and 0<v<t<, then (a del(v )b)(beta -) - K(h,2)ss o (a#,b)ss/(a del tgb)(ss) - K(h,2)(TM)-T-ss v/ (a double dagger,b)ss t(1 - t) '<br /> (3) If 0< a <b, B1 and 0<vt<1, then<br /> (a del,b)(ss) - (a!,b)B/(aV+b)-(a! b)(a,del tn)-(a,b)ss <=(a del ,b) - (a!,b) t(1-v)<br /> <br /> In addition, we obtain some new results for Young's inequality for operators.