S. S. Dragomir and Y. J. Cho. I. Introduction In 1956, J. Aczél has proved the following interesting inequality ([2, p. 57], [3, p.

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1 ON ACZÉL S INEQUALITY FOR REAL NUMBERS S. S. Dragomir ad Y. J. Cho Abstract. I this ote, we poit out some ew iequalities of Aczel s type for real umbers. I. Itroductio I 1956, J. Aczél has proved the followig iterestig iequality [, p. 57], [3, p. 117 ]): Theorem A. Let a = a 1, a,, a ) ad b = b 1, b,, b ) be two sequeces of real umbers such that The 0 < a 1 a a or 0 < b 1 b b. 1.1) a 1 a a )b 1 b b ) a 1 b 1 a b a b ), with the equality if ad oly if the sequeces a ad b are proportioal. For various geeralizatios of Theorem A, see the recet book [3, p. 117]) where further refereces are give. Now, i this ote, we give aother proof tha that embodied i [, p. 57] for a weighted variat of 1.1). Assume that p i a i a ad p i b i b, 1991) AMS Subject Classificatio: 8D15 Key Words ad Phrases: Aczél s iequality. 1 Typeset by AMS-TEX

2 S. S. DRAGOMIR AND Y. J. CHO where a i, b i, a, b R ad 0 p i for i = 1,,,. The we have the followig iequality: 1.) ) p i a i b ) ). Ideed, by a simple calculatio, we have for all a, b, c, d R. Thus we have c )b d ) ab cd ) 1.3) ) p i a i b ab ) ) 1/ ) 1/ ). p i a i By Cauchy-Buiakowski-Schwarz s iequality, we have ) 1/ p i a i p i b i ) 1/ ad so Thus, we have ) 1/ 0 ab p i a i p i b i ab = ab. ) 1/ 1.4) ab ) 1/ p i a i ) 1/ ) ).

3 ON ACZÉL S INEQUALITY FOR REAL NUMBERS 3 Therefore, from 1.3) ad 1.4), we have the iequality 1.). This completes the proof. II. The Results We will start with the followig theorem which give a refiemet of the followig variat of Aczél s iequality:.1) ) 1/ b ) 1/ p i b ab, i assumig that p i a i a ad b ad a i, b i, a, b R, 0 p i for i = 1,,,. Theorem.1. Assume that a i, b i, p i, a, b are as above ad 0 q i p i for all i = 1,,,. The we have the followig iequality:.) 0 q i a i b i [ q i a i ] 1/ q i b i ) 1/ b Proof. From p ia i 0 ad b p ia i b p i q i )a i p i q i )bi b Now, for t i = p i q i 0, by 1.), we have ) t i ai b ) t i b i p i a i 0, 0. ) 1/. p i b i 0, it follows that ), t i a i b i

4 4 S. S. DRAGOMIR AND Y. J. CHO i.e., [ ) p i a i + [ ][ q i a i b ) + ]. ) p i b i + ] q i bi Applyig the well-kow Cauchy-Buiakowski-Schwarz s iequality for real umber, we have.4) [ q i a i ) 1/ q i b i q i a i + [ ) 1/ + p i a i ) 1/ b ) 1/ ] ) q i bi + ad, by the triagle iequality, ) +.5). + ) 1/ ] p i b i [ b Thus, from.3),.4) ad.5), it follows that ) 1/ ) 1/ ) 1/ q i a i q i bi + p i a i b which implies that 0 q i a i b i ab, + ) 1/ ) 1/ q i ai q i bi This completes the proof. ) 1/ b ) 1/ ] ) p i b i ) 1/. ) 1/

5 ON ACZÉL S INEQUALITY FOR REAL NUMBERS 5 Corollary.. With the above assumptios for a i, b i, a, b R ad 0 p i for i = 1,,,, we have the followig iequality: 0 p i a i b i [ p i a i ] 1/ p i b i ) 1/ p i a i b Aother result of Aczél s type is as follows: p i b i ) 1/. Theorem.3. Assume that a, b, a i, b i R ad 0 p i for i = 1,,, are such that a ad p i b i b. The we have the followig iequality: [ a p i a i ab 1/ ) 1/ ] 1/ [ b 1/. ) 1/ ] 1/ Proof. We will start with the followig elemetary iequality:.7) x y)z u) xz yu, where x y 0 ad z u 0. Ideed, the iequality.7) is equivalet with x y)z u) xz yu) = xz xzyu + yu, i.e., xz + yu yz xu xz xzyu + yu, which is equivalet with xzyu yz + xu

6 6 S. S. DRAGOMIR AND Y. J. CHO for x, y, z, u 0, which is obvious. Now, puttig ) 1/, x = a, y = p i a i z = b, u = p i b i ) 1/, the, by the iequality.7), we have.8) [ a p i a i ) 1/ ] 1/ [ b ) 1/ ] 1/ p i b i ) 1/4 ) 1/4. ab 1/ p i a i By Cauchy-Buiakowski-Schwarz s iequality, we have 1/ [ p i a i p i b i ] 1/4 ad so, by.8), we have the desired iequality.6). This completes the proof. Corollary.4. Let a, b, a i, b i R for i = 1,,, be such that a i a ad b i b. The we have the followig iequality: [ a a i ) 1/ ] 1/ [ b ab 1/ 1/ a i b i. b i ) 1/ ] 1/ Remark. The iequality.9) was proved i [1] as a particular case of a iequality holdig i ier product spaces.

7 ON ACZÉL S INEQUALITY FOR REAL NUMBERS 7 Refereces 1. S. S. Dragomir, A geeralizatio of J. Aczél s iequality i ier product spaces, Acta Math. Hugarica 65) 1994), D. S. Mitriović, Aalytic Iequalities, Spriger-Verlag, D. S. Mitriović, J. E. Pečarić ad A. M. Fik, Classical ad New Iequalities i Aalysis, Kluwer Acad. Publ., Departmet of Mathematics, Victoria Uiversity of Techology, Melboure, Victoria 8001, Australia Departmet of Mathematics, Gyeogsag Natioal Uiversity, Chiju , Korea