OPTIMAL GENERALIZED LOGARITHMIC MEAN BOUNDS FOR THE GEOMETRIC COMBINATION OF ARITHMETIC AND HARMONIC MEANS | Long | Journal of the Indonesian Mathematical Society 5 391 1 PB

(1)

J. Indones. Math. Soc.

Vol. 17, No. 2 (2011), pp. 85–95.

OPTIMAL GENERALIZED LOGARITHMIC MEAN

BOUNDS FOR THE GEOMETRIC COMBINATION OF

ARITHMETIC AND HARMONIC MEANS

Bo-Yong Long

1

and Yu-Ming Chu

2

1

_{College of Mathematics Science, Anhui University, Hefei 230039, P. R.}

China, [email protected]

2

_{Department of Mathematics, Huzhou Teachers College, Huzhou 313000, P.}

R. China, [email protected]

Abstract. _{In this paper, we answer the question: for}_α _∈ _(0,_{1), what are the}

greatest valuep=p(α) and least valueq=q(α), such that the double inequality Lp(a, b) ≤Aα(a, b)H1−α(a, b)≤ Lq(a, b) holds for all a, b > 0? where Lp(a, b),

A(a, b), andH(a, b) are thep-th generalized logarithmic, arithmetic, and harmonic means ofaandb, respectively.

Key words: Generalized logarithmic mean, arithmetic mean, harmonic mean.

2000 Mathematics Subject Classification: 26E60.