OPTIMAL INEQUALITIES RELATED TO THE LOGARITHMIC, IDENTRIC, ARITHMETIC AND HARMONIC MEANS

. The logarithmic mean L ( a, b ), identric mean I ( a, b ), arithmetic mean A ( a, b ) and harmonic mean H ( a, b ) of two positive real values a and b are deﬁned by

respectively.In the recent past, both mean values have been the subject of intensive research.In particular, many remarkable inequalities for logarithmic mean or identric mean can be found in the literature .It might be surprising that the logarithmic mean has applications in physics, economics, and even in meteorology [26][27][28].In [26] the authors study a variant of Jensen's functional equation involving L(a, b), which appears in a heat conduction problem.
The power mean M p (a, b) of order p is defined by a+b the arithmetic mean, geometric mean and harmonic mean of two positive numbers a and b, respectively, then it is well-known that and all inequalities are strict for a = b.
In [9,12,30]  The following companion of (1.3) provides inequalities for the geometric and arithmetic means of L(a, b) and I(a, b).A proof can be found in [7]. in terms of power means are proved in [4, 6-8, 10, 24, 30].
for all a, b > 0 with a = b.

MAIN RESULTS
Theorem 3.1.The double inequality holds for all a, b > 0 if and only if α 1 ≤ 0 and ) follows from Lemma 2.1 and (3.1).Secondly, we prove that the parameters α 1 ≤ 0 and β 1 ≥ 2 3 cannot be improved.
The logarithmic mean L(a, b) and identric mean I(a, b) of two positive real values a and b are defined by (

2 3 1 3
the authors present bounds for L(a, b) in terms of G(a, b) and A(a, b).G (a, b)A (a, b) < L(a, b) < 2 3 G(a, b) + 1 3 A(a, b) for all a, b > 0 with a = b.

1 2
, b) for all a, b > 0 with a = b.The following bounds for L(a, b), I(a, b), (L(a, b)I(a, b)) , and L(a,b)+I(a,b) 2

2 3
we assume that a = b.Firstly, we prove that H(a, b) < L(a, b) < 2 3 A(a, b) + 1 3 H(a, b).From (1.3) we know that H(a, b) < L(a, b) is true, so we only need to prove that L(a, b) < A(a, b) + 1 3 H(a, b).Without loss of generality, we assume that a > b.Let t = a b > 1, then simple computation leads to 2 3 A(a, b)

1 6 H
(a, b) > I(a, b) for a, b > 0 with a = b.