Zagreb University Zagreb

ABSTRACT. An extension is given for the inverse to Hhlder’s inequality obtained recently by Zhuang. KEY WORDS AND PHRASES. Inverse Hhlder inequality. 1991 AMS SUBJECT CLASSIFICATION CODE. 26D15. Recently Zhuang [1] proved the following inverse of the arithmetico-geometric inequality. THEOREMA. Let O<a<_x<_A,O<b<_y<B, lip+l/q = 1, p> 1;then x y fA/p+b/q a/p+B/q+ _< max [ i7, a’i,B’i, f x’i’y’i (1)P q or x + y < _ max A1/pbl/q ai./--i-/q xl/’yi/q, (2) the sign of equality in (1) and (2) holds if and only if either (x,y) (a,B) or (x,y) (A,b). Moreover, if a _>/5’, then alp + B/q xl/,yl/q < _ x y Alp + b/q zi/pyi/q+- < (3)alh’B1/q p q- Ai/,bi/q the sign of equality on the right-hand side of (3) holds if and only if (x,y) (A,b), and the sign of equality on the left-hand side of () holds if and only if (.x,y) (a, B). The sign of inequality in (3) is reversed if b _> A. This enables us to formulate the following theorem