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Some new inequalities for means in two variables. (English) Zbl 1154.26029

The following results are given simple proofs using a lemma that states that if f,g satisfy the usual conditions for the mean-value theorem and if f ' /g ' is increasing so are the ratios f ( x ) - f ( b )/g ( x ) - g ( b ),f ( x ) - f ( a )/g ( x ) - g ( a ).

If p1 then: α p A p +(1-α p )G p <L p <β p A p +(1-β p )G p α p 0β p 1/3;

if 0p6/5 then: α p A p +(1-α p )G p <I p <β p A p +(1-β p )G p α p 2/3β p (2/e) p ;

if p2 then: α p A p +(1-α p )G p <I p <β p A p +(1-β p )G p α p (2/e) p β p 2/3.

A,G,L,I are the arithmetic, geometric, logarithmic and identric means of two variables respectively. These results extend earlier inequalities of Trif, Sandor and Trif, Alzer and Qui, Zhu and Wu.

MSC:
26E60Means
26D07Inequalities involving other types of real functions