Abstract
We find the greatest valuepand the least valueqin(0,1/2)such that the double inequalityH(pa+(1-p)b,pb+(1-p)a)<I(a,b)<H(qa+(1-q)b,qb+(1-q)a)holds for alla,b>0witha≠b. Here,H(a,b), andI(a,b)denote the harmonic and identric means of two positive numbersaandb, respectively.
Highlights
The classical harmonic mean H a, b and identric mean I a, b of two positive numbers a and b are defined by H a, b 2ab ab ⎧ I a, b⎪⎪⎨ 1 ⎪⎪⎩ae, bb aa 1/ b−a, a / b, a b, 1.2 respectively
We find the greatest value p and the least value q in 0, 1/2 such that the double inequality H pa 1 − p b, pb 1 − p a < I a, b < H qa 1 − q b, qb 1 − q a holds for all a, b > 0 with a / b
Both mean values have been the subject of intensive research
Summary
We find the greatest value p and the least value q in 0, 1/2 such that the double inequality H pa 1 − p b, pb 1 − p a < I a, b < H qa 1 − q b, qb 1 − q a holds for all a, b > 0 with a / b. It is natural to ask what are the greatest value p and the least value q in 0, 1/2 such that the double inequality H pa 1 − p b, pb 1 − p a < I a, b < H qa 1 − q b, qb 1 − q a holds for all a, b > 0 with a / b.
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